Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1

*Zhou Guo-Quan*

## **Abstract**

Based upon different methods such as a newly revised version of inverse scattering transform, Marchenko formalism, and Hirota's bilinear derivative transform, this chapter aims to study and solve the derivative nonlinear Schrödinger (DNLS for brevity) equation under vanishing boundary condition (VBC for brevity). The explicit one-soliton and multi-soliton solutions had been derived by some algebra techniques for the VBC case. Meanwhile, the asymptotic behaviors of those multi-soliton solutions had been analyzed and discussed in detail.

**Keywords:** soliton, nonlinear equation, derivative nonlinear Schrödinger equation, inverse scattering transform, Zakharov-Shabat equation, Marchenko formalism, Hirota's bilinear derivative transform, rogue wave

## **1. Introduction**

Derivative nonlinear Schrödinger (DNLS for brevity) equation is one of the several rare kinds of integrable nonlinear models. Research of DNLS equation has not only mathematic interest and significance, but also important physical application background. It was first found that the Alfven waves in space plasma [1–3] can be modeled with DNLS equation. The modified nonlinear Schrödinger (MNLS for brevity) equation, which is used to describe the sub-picosecond pulses in single mode optical fibers [4–6], is actually a transformed version of DNLS equation. The weak nonlinear electromagnetic waves in ferromagnetic, anti-ferromagnetic, or dielectric systems [5–9] under external magnetic fields can also be modeled by DNLS equation.

Although DNLS equation is similar to NLS equation in form, it does not belong to the famous AKNS hierarchy at all. As is well known, a nonlinear integrable equation can be transformed to a pair of Lax equation satisfied by its Jost functions, the original nonlinear equation is only the compatibility condition of the Lax pair, that is, the so-called zero-curvature condition. Another fact had been found by some scholars that those nonlinear integrable equations which have the same first operator of the Lax pair belong to the same hierarchy and can deal with the same inverse scattering transform (IST for brevity). As a matter of fact, the DNLS

equation has a squared spectral parameter of *λ*<sup>2</sup> in the first operator of its Lax pair, while the famous NLS equation, one typical example in AKNS hierarchy, has a spectral parameter of *λ*. Thus, the IST of the DNLS equation is greatly different from that of the NLS equation which is familiar to us. In a word, it deserves us to demonstrate several different approaches of solving it as a typical integrable nonlinear equation.

**2.1 The revised inverse scattering transform and the Zakharov-Shabat equation**

DNLS equation for the one-dimension wave function *u x*ð Þ , *t* is usually

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

*iut* <sup>þ</sup> *uxx* <sup>þ</sup> *i u*j j<sup>2</sup>

*<sup>U</sup>* � *<sup>i</sup>λ*<sup>2</sup>

1 0 � �*<sup>e</sup>*

The Jost solutions of (4) are defined by their asymptotic behaviors as *x* ! �∞.

where *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>* ð Þ *<sup>ψ</sup>*1ð Þ *<sup>x</sup>*, *<sup>λ</sup>* , *<sup>ψ</sup>*2ð Þ *<sup>x</sup>*, *<sup>λ</sup>* T, *<sup>ψ</sup>*~ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>* ð Þ *<sup>ψ</sup>*~1ð Þ *<sup>x</sup>*, *<sup>λ</sup>* , *<sup>ψ</sup>*~2ð Þ *<sup>x</sup>*, *<sup>λ</sup>* T, etc., and

Since the first Lax equation of DNLS is similar to that of NLS, there are some similar properties of the Jost solutions. The monodromy matrix Tð Þ*λ* is defined as

superscript "T" represents transposing of a matrix here and afterwards.

<sup>T</sup>ð Þ¼ *<sup>λ</sup> a t*ð Þ� , *<sup>λ</sup>* <sup>~</sup>

*u* � �

with VBC, where the subscripts stand for partial derivative. Eq. (1) is also called

*<sup>σ</sup>*<sup>3</sup> <sup>þ</sup> *<sup>λ</sup>U*, *<sup>U</sup>* <sup>¼</sup> <sup>0</sup> *<sup>u</sup>*

*U*2

where *λ* is a spectral parameter, and *σ*<sup>3</sup> is the third one of Pauli matrices *σ*1, *σ*2, *σ*3, and a bar over a letter, (e.g., *u* in (2)), represents complex conjugate. The first Lax

�*u* 0

*<sup>σ</sup>*<sup>3</sup> � *<sup>λ</sup>* �*U*<sup>3</sup> <sup>þ</sup> *iUxσ*<sup>3</sup>

*<sup>∂</sup><sup>x</sup> f x*ð Þ¼ , *<sup>λ</sup> L x*ð Þ , *<sup>λ</sup> f x*ð Þ , *<sup>λ</sup>* (4)

�*iλ*2*<sup>x</sup>*, *<sup>E</sup>*•2ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>*

Ψð Þ¼ *x*, *λ* ð*ψ*~ð Þ *x*, *λ* , *ψ*ð Þ *x*, *λ* Þ ! *E x*ð Þ , *λ* , as *x* ! ∞ (7) <sup>Φ</sup>ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup> <sup>ϕ</sup>*ð Þ *<sup>x</sup>*, *<sup>λ</sup>* , *<sup>ϕ</sup>*~ð Þ *<sup>x</sup>*, *<sup>λ</sup>* � � ! *E x*ð Þ , *<sup>λ</sup>* , as *<sup>x</sup>* ! �<sup>∞</sup> (8)

Φð Þ¼ *x*, *λ* Ψð Þ *x*, *λ* Tð Þ*λ* , (9)

*b t*ð Þ , *λ*

*b t*ð Þ , *λ a t* ~ð Þ , *λ*

!

*<sup>σ</sup>*3; *<sup>M</sup>* ! *<sup>M</sup>*<sup>0</sup> ¼ �*i*2*λ*<sup>4</sup>*σ*<sup>3</sup> (5)

0 1 � �*<sup>e</sup>*

*<sup>i</sup>λ*2*<sup>x</sup>* (6)

(10)

*<sup>x</sup>* <sup>¼</sup> <sup>0</sup> (1)

� � (2)

� � (3)

*2.1.1 The fundamental concepts for the IST theory of DNLS equation*

Kaup-Newell (KN for brevity) equation. Its Lax pair is given by

*<sup>L</sup>* ¼ �*iλ*<sup>2</sup>

*<sup>L</sup>* ! *<sup>L</sup>*<sup>0</sup> ¼ �*iλ*<sup>2</sup>

�*iλ*2*xσ*<sup>3</sup> ; *<sup>E</sup>*•1ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>*

*<sup>M</sup>* ¼ �*i*2*λ*<sup>4</sup>*σ*<sup>3</sup> <sup>þ</sup> <sup>2</sup>*λ*<sup>3</sup>

In the limit of ∣*x*∣ ! ∞, *u* ! 0, and

The free Jost solution is a 2 � 2 matrix.

*E x*ð Þ¼ , *λ e*

**for DNLS equation with VBC**

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

expressed as

and

equation is

where

**25**

In this chapter, we will solve the DNLS equation under two kinds of boundary condition, that is, the vanishing boundary condition (VBC for brevity) and the nonvanishing boundary condition (NVBC for brevity), by means of three different methods – the revised IST method, the Marchenko formalism, and the Hirota's bilinear derivative method. Meanwhile, we will search for different types of special soliton solution to the DNLS equation, such as the light/dark solitons, the pure solitons, the breather-type solitons, and the rogue wave solution, in one- or multisoliton form.

## **2. An N-soliton solution to the DNLS equation based on a revised inverse scattering transform**

For the VBC case of DNLS equation, which is just the concerned theme of the section, some attempts and progress have been made to solve the DNLS equation. Since Kaup and Newell proposed an IST with a revision in their pioneer works [10, 11], one-soliton solution was firstly attained and several versions of raw or explicit multi-soliton solutions were also obtained by means of different approaches [12–20]. Huang and Chen have got a *N*� soliton solution by means of Darboux transformation [15]. Steudel has derived a formula for *N*� soliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation [13]; but just as Chen points out in Ref. [16], Steudel's multi-soliton solution is difficult to demonstrate collisions among solitons and still has a too complicate form to be used in the soliton perturbation theory of DNLS equation, although it can easily generate compute pictures. Since the integral kernel in Zakharov-Shabat (Z-S for brevity) equation does not tend to zero in the limit of spectral parameter *λ* with j j *λ* ! ∞, the contribution of the path integral along the big circle (the out contour) is also nonvanishing, the usual procedure to perform inverse scattering transform encounters difficulty and is invalid. Kaup thus proposed a revised IST by multiplying an additional weighing factor before the Jost solution *E x*ð Þ , *λ* , so that it tends to zero as ∣*λ*∣ ! ∞, thus the modified *Z*-*S* kernel should lead to vanishing contribution of the integral along the big circle of Cauchy contour. Though the one-soliton solution has been found by the obtained Z-S equation of their IST, it is very difficult to derive directly its multi-soliton solution by their IST due to the existence of a complicated phase factor which is related to the solution itself [11]. We thus consider proposing a new revised IST to avoid the excessive complexity. Our *N***-**soliton solution obviously has a standard multi-soliton form. It can be easily used to discuss its asymptotic behaviors and then develop its direct perturbation theory. On the other hand, in solving Z-S equation for DNLS with VBC, unavoidably we will encounter a problem of calculating determinant det I þ Q1Q2 ð Þ, for two *N* � *N* matrices Q1 and Q2, where I is a *N* � *N* identity matrix. Our work also shows Binet-Cauchy formula and some other linear algebra techniques, (**Appendices** A.1–4 in Part 2), play important roles in the whole process, and actually also effective for some other nonlinear integrable models [21].

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

## **2.1 The revised inverse scattering transform and the Zakharov-Shabat equation for DNLS equation with VBC**

### *2.1.1 The fundamental concepts for the IST theory of DNLS equation*

DNLS equation for the one-dimension wave function *u x*ð Þ , *t* is usually expressed as

$$\left(\dot{u}\_t + u\_{\text{xx}} + i\left(\left|u\right|^2 u\right)\_{\text{x}} = \mathbf{0} \tag{1}$$

with VBC, where the subscripts stand for partial derivative. Eq. (1) is also called Kaup-Newell (KN for brevity) equation. Its Lax pair is given by

$$L = -i\lambda^2 \sigma\_3 + \lambda U,\\ U = \begin{pmatrix} \mathbf{0} & u \\ -\overline{u} & \mathbf{0} \end{pmatrix} \tag{2}$$

and

$$M = -i2\lambda^4 \sigma\_3 + 2\lambda^3 U - i\lambda^2 U^2 \sigma\_3 - \lambda \left(-U^3 + iU\_\ge \sigma\_3\right) \tag{3}$$

where *λ* is a spectral parameter, and *σ*<sup>3</sup> is the third one of Pauli matrices *σ*1, *σ*2, *σ*3, and a bar over a letter, (e.g., *u* in (2)), represents complex conjugate. The first Lax equation is

$$
\partial\_{\mathbf{x}} f(\mathbf{x}, \ \lambda) = L(\mathbf{x}, \ \lambda) f(\mathbf{x}, \ \lambda) \tag{4}
$$

In the limit of ∣*x*∣ ! ∞, *u* ! 0, and

$$L \to L\_0 = -i\lambda^2 \sigma\_3;\ M \to M\_0 = -i2\lambda^4 \sigma\_3 \tag{5}$$

The free Jost solution is a 2 � 2 matrix.

$$E(\mathbf{x},\ \lambda) = e^{-i\hat{\lambda}^2 \mathbf{x} \sigma\_3}; E\_{\mathbf{1}}(\mathbf{x}, \lambda) = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix} e^{-i\hat{\lambda}^2 \mathbf{x}}, E\_{\mathbf{2}}(\mathbf{x}, \lambda) = \begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix} e^{i\hat{\lambda}^2 \mathbf{x}} \tag{6}$$

The Jost solutions of (4) are defined by their asymptotic behaviors as *x* ! �∞.

$$\Psi(\mathbf{x},\boldsymbol{\lambda}) = (\tilde{\boldsymbol{\mu}}(\mathbf{x},\boldsymbol{\lambda}), \boldsymbol{\mu}(\mathbf{x},\boldsymbol{\lambda})) \to E(\mathbf{x},\boldsymbol{\lambda}), \mathbf{as} \,\boldsymbol{\mathfrak{x}} \to \infty \tag{7}$$

$$\Phi(\mathbf{x},\lambda) = \left(\phi(\mathbf{x},\lambda), \tilde{\phi}(\mathbf{x},\lambda)\right) \to E(\mathbf{x},\lambda), \mathbf{as}\,\mathbf{x} \to -\infty \tag{8}$$

where *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>* ð Þ *<sup>ψ</sup>*1ð Þ *<sup>x</sup>*, *<sup>λ</sup>* , *<sup>ψ</sup>*2ð Þ *<sup>x</sup>*, *<sup>λ</sup>* T, *<sup>ψ</sup>*~ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>* ð Þ *<sup>ψ</sup>*~1ð Þ *<sup>x</sup>*, *<sup>λ</sup>* , *<sup>ψ</sup>*~2ð Þ *<sup>x</sup>*, *<sup>λ</sup>* T, etc., and superscript "T" represents transposing of a matrix here and afterwards.

Since the first Lax equation of DNLS is similar to that of NLS, there are some similar properties of the Jost solutions. The monodromy matrix Tð Þ*λ* is defined as

$$\Phi(\mathbf{x}, \boldsymbol{\lambda}) = \Psi(\mathbf{x}, \boldsymbol{\lambda}) \cdot \mathbf{T}(\boldsymbol{\lambda}), \tag{9}$$

where

$$\mathbf{T}(\boldsymbol{\lambda}) = \begin{pmatrix} a(t, \boldsymbol{\lambda}) & -\tilde{b}(t, \boldsymbol{\lambda}) \\ b(t, \boldsymbol{\lambda}) & \tilde{a}(t, \boldsymbol{\lambda}) \end{pmatrix} \tag{10}$$

It is easy to find from (2) and (9) that

$$
\sigma\_2 \ L(\overline{\lambda}) \ \sigma\_2 = L(\lambda), \ \sigma\_2 \ T(\overline{\lambda}) \ \sigma\_2 = T(\lambda) \tag{11}
$$

Eq. (21) leads to *gxυ*<sup>1</sup> ¼ *λuυ*2. Considering (25), in the limit of j j *λ* ! ∞, we find a

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

which expresses the conjugate of solution *u* in terms of the Jost solutions as

*<sup>a</sup>*ð Þ¼ *<sup>λ</sup>* <sup>Y</sup> *N*

> 2 *n=λ*<sup>2</sup>

Due to *μ*<sup>0</sup> 6¼ 0 in (24) and (25), the Jost solutions do not tend to free Jost solutions *E x*ð Þ , *λ* in the limit of ∣*λ*∣ ! ∞. This is their most typical property which means that the usual procedure of constructing the equation of IST by a Cauchy contour integral must be invalid and abortive, thus a newly revised procedure to derive a suitable IST and the corresponding Z-S equation is proposed in our group.

*2.1.3 The revised IST and Zakharov-Shabat equation for DNLS equation with VBC*

<sup>Θ</sup>ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup> <sup>ϕ</sup>*ð Þ *<sup>x</sup>*, *<sup>λ</sup> <sup>=</sup>a*ð Þ*<sup>λ</sup>* , as *<sup>λ</sup>* in I, III quadrants*: ψ*~ð Þ *x*, *λ* , as *λ* in II, IV quadrants*:*

> <sup>d</sup>*λ*<sup>0</sup> <sup>1</sup> *λ*<sup>0</sup> � *λ*

∣*λ*∣!∞

be chosen as shown in **Figure 1**, where the radius of big circle tends to infinite, while the radius of small circle tends to zero. And the factor *λ*�<sup>2</sup> is introduced to ensure the contribution of the integral along the big arc is vanishing. Meanwhile,

1

*<sup>e</sup><sup>i</sup>λ*2*<sup>x</sup>* <sup>¼</sup> 0,

The 2 � 1 column function Θð Þ *x*, *λ* can be introduced as usual

�

An alternative form of IST equation is proposed as

*<sup>i</sup>λ*2*<sup>x</sup>* <sup>¼</sup> <sup>1</sup> 2*πi* ð Γ

*<sup>x</sup>*>0, Im *<sup>λ</sup>*<sup>2</sup> <sup>&</sup>gt;0, ð Þ *<sup>λ</sup>* in the I, III quadrants , *<sup>x</sup>*<0, Im *<sup>λ</sup>*<sup>2</sup> <sup>&</sup>lt;0, ð Þ *<sup>λ</sup>* in the II, IV quadrants ,

<sup>d</sup>*λ*<sup>0</sup> ln *<sup>a</sup> <sup>λ</sup>*<sup>0</sup> ð Þ*a*<sup>~</sup> *<sup>λ</sup>*<sup>0</sup> ð Þ

*n*¼1

where *a*ð Þ¼ 0 1. It comes from our consideration of the fact that, from the sum

in order to maintain that ln *a*ð Þ!*λ* 0, as *λ* ! 0, and ln *a*ð Þ*λ* is finite as j j *λ* ! ∞,

*ϕ*ð Þ¼ *x*, *λ<sup>n</sup> bnψ*ð Þ *x*, *λ<sup>n</sup>* , *a*\_ð Þ¼� �*λ<sup>n</sup> a*\_ð Þ *λ<sup>n</sup>* , *bn*þ*<sup>N</sup>* ¼ �*bn* (28)

On the other hand, the zeros of *a*ð Þ*λ* appear in pairs and can be designed by *λn*, *n* ¼ 1, 2, ⋯, *N* in the I quadrant, and *λ<sup>n</sup>*þ*<sup>N</sup>* ¼ �*λ<sup>n</sup>* in the III quadrant. The discrete

> *<sup>λ</sup>*<sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> *n <sup>λ</sup>*<sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> *n* • *λ*2 *n λ*2 *n*

*<sup>λ</sup>*<sup>0</sup> *<sup>λ</sup>*<sup>0</sup> ð Þ � *<sup>λ</sup>* , <sup>Γ</sup> <sup>¼</sup> ð Þ 0, <sup>∞</sup> <sup>∪</sup>ð Þ *<sup>i</sup>*∞, *<sup>i</sup>*<sup>0</sup> <sup>∪</sup>ð Þ 0, �<sup>∞</sup> <sup>∪</sup>ð Þ �*i*∞, *<sup>i</sup>*<sup>0</sup> ,

*<sup>n</sup>* in (27). At the zeros of *a*ð Þ*λ* , we have

*<sup>λ</sup>*0<sup>2</sup> <sup>Θ</sup><sup>1</sup> *<sup>x</sup>*, *<sup>λ</sup>*<sup>0</sup> ð Þ� *<sup>E</sup>*11ð*x*, *<sup>λ</sup>*<sup>0</sup> f gÞ *<sup>e</sup>*

then the integral path Γ should

*λψ*~2ð Þ *x*, *λ =ψ*~1ð Þ *x*, *λ* (26)

(27)

(29)

*<sup>x</sup>* (30)

*iλ*0<sup>2</sup>

*u* ¼ *i*2 lim ∣*λ*∣!∞

useful formula

part of *a*ð Þ*λ* is [21–23].

of two Cauchy integrals

*<sup>λ</sup>* <sup>þ</sup> <sup>0</sup> <sup>¼</sup> <sup>1</sup>

2*πi* ð Γ

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

we then have to introduce a factor *λ*

ln *a*ð Þ*λ*

1

as

**27**

(

*<sup>λ</sup>*<sup>2</sup> f g <sup>Θ</sup>1ð Þ� *<sup>x</sup>*, *<sup>λ</sup> <sup>E</sup>*11ð*x*, *<sup>λ</sup>*<sup>Þ</sup> *<sup>e</sup>*

Because in the limit of j j *λ* ! ∞, lim

j j *λ* ! ∞.

$$
\sigma\_2 \, \overline{\Psi(\mathbf{x}, \overline{\lambda})} \, \sigma\_2 = \Psi(\mathbf{x}, \lambda), \, \sigma\_2 \, \overline{\Phi(\mathbf{x}, \overline{\lambda})} \, \, \sigma\_2 = \Phi(\mathbf{x}, \lambda) \tag{12}
$$

and

$$
\sigma\_3 \Psi(\mathbf{x}, \boldsymbol{\lambda}) \sigma\_3 = \Psi(\mathbf{x}, -\boldsymbol{\lambda}), \ \sigma\_3 \Phi(\mathbf{x}, \boldsymbol{\lambda}) \sigma\_3 = \Phi(\mathbf{x}, -\boldsymbol{\lambda}) \tag{13}
$$

$$
\sigma\_3 \mathbf{L}(\lambda) \sigma\_3 = \mathbf{L}(-\lambda), \ \sigma\_3 \mathbf{T}(\lambda) \sigma\_3 = \mathbf{T}(-\lambda) \tag{14}
$$

Then we can get the following reduction relation and symmetry properties

$$i\sigma\_2 \overline{\hat{\psi}(\mathfrak{x}, \overline{\hat{\lambda}})} = \tilde{\psi}(\mathfrak{x}, \lambda) \tag{15}$$

$$-i\sigma\_2 \overline{\rho}(\mathbf{x}, \overline{\lambda}) = \tilde{\rho}(\mathbf{x}, \lambda) \tag{16}$$

$$
\overline{\tilde{a}}\left(\overline{\lambda}\right) = a(\lambda); \quad \overline{\tilde{b}}\left(\overline{\lambda}\right) = b(\lambda) \tag{17}
$$

and

$$
\psi(\mathbf{x}, -\boldsymbol{\lambda}) = -\sigma\_3 \psi(\mathbf{x}, \boldsymbol{\lambda})\tag{18}
$$

$$
\tilde{\boldsymbol{\varphi}}(\mathbf{x}, -\boldsymbol{\lambda}) = \sigma\_3 \tilde{\boldsymbol{\varphi}}(\mathbf{x}, \boldsymbol{\lambda}) \tag{19}
$$

$$\begin{aligned} a(-\lambda) &= a(\lambda); \; b(-\lambda) = -b(\lambda) \\ \tilde{a}(-\lambda) &= \tilde{a}(\lambda); \; \tilde{b}(-\lambda) = -\tilde{b}(\lambda) \end{aligned} \tag{20}$$

### *2.1.2 Relation between Jost functions and the solutions to the DNLS equation*

The asymptotic behaviors of the Jost solutions in the limit of ∣*λ*∣ ! ∞ can be obtained by simple derivation. Let *υ* ¼ ð Þ *υ*1, *υ*<sup>2</sup> *<sup>T</sup>* � *<sup>ψ</sup>*~ð Þ *<sup>x</sup>*, *<sup>λ</sup>* ; Eq. (4) can be rewritten as

$$
\nu\_{1\mathbf{x}} + i\lambda^2 \nu\_1 = \lambda\mu\nu\_2,\\
\nu\_{2\mathbf{x}} - i\lambda^2 \nu\_2 = -\lambda\overline{\mu}\nu\_1 \tag{21}
$$

Then we have

$$(\nu\_{1\text{xx}} - \mu\_{\text{x}}(\nu\_{1\text{x}} + i\lambda^2 \nu\_1)/u + \lambda^4 \nu\_1 + \lambda^2 |u|^2 \nu\_1 = 0 \tag{22}$$

In the limit <sup>∣</sup>*λ*<sup>∣</sup> ! <sup>∞</sup>, we assume *<sup>ψ</sup>*~1ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup> <sup>e</sup>*�*iλ*2*x*þ*<sup>g</sup>* , substituting it into Eq. (22), then we have

$$\left(-i\lambda^2 + \mathbf{g}\_x\right)^2 + \mathbf{g}\_{xx} - u\_x \mathbf{g}\_x / u + \lambda^4 + \lambda^2 |u|^2 = \mathbf{0} \tag{23}$$

In the limit <sup>∣</sup>*λ*<sup>∣</sup> ! <sup>∞</sup>, *gx* can be expanded as series of *<sup>λ</sup>*�<sup>2</sup> *<sup>j</sup>* , *j* ¼ 1, 2, ⋯.

$$\text{sig}\_x \equiv \mu = \mu\_0 + \mu\_2 \left( 2\lambda^2 \right)^{-1} + \dotsb \tag{24}$$

and

$$
\mu\_0 = |u|^2/2, \mu\_2 = -i\overline{u}\_x u/2 - |u|^4/4, \dots \tag{25}
$$

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

Eq. (21) leads to *gxυ*<sup>1</sup> ¼ *λuυ*2. Considering (25), in the limit of j j *λ* ! ∞, we find a useful formula

$$\overline{u} = i2 \lim\_{|\lambda| \to \infty} \lambda \tilde{\nu}\_2(\infty, \lambda) / \tilde{\nu}\_1(\infty, \lambda) \tag{26}$$

which expresses the conjugate of solution *u* in terms of the Jost solutions as j j *λ* ! ∞.

On the other hand, the zeros of *a*ð Þ*λ* appear in pairs and can be designed by *λn*, *n* ¼ 1, 2, ⋯, *N* in the I quadrant, and *λ<sup>n</sup>*þ*<sup>N</sup>* ¼ �*λ<sup>n</sup>* in the III quadrant. The discrete part of *a*ð Þ*λ* is [21–23].

$$a(\lambda) = \prod\_{n=1}^{N} \frac{\lambda^2 - \lambda\_n^2}{\lambda^2 - \overline{\lambda\_n^2}} \bullet \frac{\overline{\lambda\_n^2}}{\lambda\_n^2} \tag{27}$$

where *a*ð Þ¼ 0 1. It comes from our consideration of the fact that, from the sum of two Cauchy integrals

$$\frac{\ln a(\boldsymbol{\lambda})}{\boldsymbol{\lambda}} + \mathbf{0} = \frac{1}{2\pi i} \int\_{\Gamma} \mathbf{d}\boldsymbol{\lambda}' \frac{\ln a(\boldsymbol{\lambda}')\tilde{a}(\boldsymbol{\lambda}')}{\boldsymbol{\lambda}'(\boldsymbol{\lambda}' - \boldsymbol{\lambda})}, \\ \boldsymbol{\Gamma} = (\mathbf{0}, \infty) \cup (i\infty, i\mathbf{0}) \cup (\mathbf{0}, -\infty) \cup (-i\infty, i\mathbf{0}),$$

in order to maintain that ln *a*ð Þ!*λ* 0, as *λ* ! 0, and ln *a*ð Þ*λ* is finite as j j *λ* ! ∞, we then have to introduce a factor *λ* 2 *n=λ*<sup>2</sup> *<sup>n</sup>* in (27). At the zeros of *a*ð Þ*λ* , we have

$$
\phi(\mathbf{x}, \boldsymbol{\lambda}\_n) = b\_n \boldsymbol{\mu}(\mathbf{x}, \boldsymbol{\lambda}\_n), \\
\dot{\mathbf{d}}(-\boldsymbol{\lambda}\_n) = -\dot{\mathbf{d}}(\boldsymbol{\lambda}\_n), \\
\mathbf{b}\_{n+N} = -\boldsymbol{b}\_n \tag{28}
$$

Due to *μ*<sup>0</sup> 6¼ 0 in (24) and (25), the Jost solutions do not tend to free Jost solutions *E x*ð Þ , *λ* in the limit of ∣*λ*∣ ! ∞. This is their most typical property which means that the usual procedure of constructing the equation of IST by a Cauchy contour integral must be invalid and abortive, thus a newly revised procedure to derive a suitable IST and the corresponding Z-S equation is proposed in our group.

#### *2.1.3 The revised IST and Zakharov-Shabat equation for DNLS equation with VBC*

The 2 � 1 column function Θð Þ *x*, *λ* can be introduced as usual

$$\Theta(\mathbf{x}, \boldsymbol{\lambda}) = \begin{cases} \phi(\mathbf{x}, \boldsymbol{\lambda}) / a(\boldsymbol{\lambda}), & \text{as } \boldsymbol{\lambda} \text{ in } \mathbf{I}, \text{ III quadrants.} \\ \boldsymbol{\tilde{\psi}}(\mathbf{x}, \boldsymbol{\lambda}), & \text{as } \boldsymbol{\lambda} \text{ in } \mathbf{II}, \text{ IV quadrants.} \end{cases} \tag{29}$$

An alternative form of IST equation is proposed as

$$\frac{1}{\lambda^2} \{\Theta\_1(\mathbf{x}, \boldsymbol{\lambda}) - E\_{11}(\mathbf{x}, \boldsymbol{\lambda})\} e^{i\boldsymbol{\hat{\lambda}'^{\mathbf{x}}\mathbf{x}}} = \frac{1}{2\pi i} \int\_{\Gamma} \mathbf{d}\boldsymbol{\lambda}' \frac{1}{\boldsymbol{\hat{\lambda}'} - \boldsymbol{\hat{\lambda}}} \frac{1}{\boldsymbol{\hat{\lambda}'^2}} \{\Theta\_1(\mathbf{x}, \boldsymbol{\lambda}') - E\_{11}(\mathbf{x}, \boldsymbol{\lambda}')\} e^{i\boldsymbol{\hat{\lambda}'^{\mathbf{x}}\mathbf{x}}} \tag{30}$$

Because in the limit of j j *λ* ! ∞, lim ∣*λ*∣!∞ *<sup>e</sup><sup>i</sup>λ*2*<sup>x</sup>* <sup>¼</sup> 0,

as *<sup>x</sup>*>0, Im *<sup>λ</sup>*<sup>2</sup> <sup>&</sup>gt;0, ð Þ *<sup>λ</sup>* in the I, III quadrants , *<sup>x</sup>*<0, Im *<sup>λ</sup>*<sup>2</sup> <sup>&</sup>lt;0, ð Þ *<sup>λ</sup>* in the II, IV quadrants , ( then the integral path Γ should

be chosen as shown in **Figure 1**, where the radius of big circle tends to infinite, while the radius of small circle tends to zero. And the factor *λ*�<sup>2</sup> is introduced to ensure the contribution of the integral along the big arc is vanishing. Meanwhile,

where

*UN* <sup>¼</sup> <sup>X</sup> *N*

*VN* <sup>¼</sup> <sup>1</sup> <sup>þ</sup><sup>X</sup>

the symmetry and reduction relation (15), we can attain

�*iλ*<sup>2</sup>

2*λ*<sup>2</sup> *m λ*2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*

*mx* <sup>þ</sup><sup>X</sup> *N*

*<sup>n</sup>* <sup>¼</sup> ffiffiffiffiffiffi 2*cn*

*f <sup>n</sup>*,ð Þ *B*<sup>2</sup> *mn* ¼ *f <sup>m</sup>*

ð Þ *<sup>w</sup>*<sup>2</sup> *<sup>m</sup>* <sup>¼</sup> *<sup>f</sup> <sup>m</sup>* <sup>þ</sup><sup>X</sup>

ð Þ *<sup>w</sup>*<sup>1</sup> *<sup>m</sup>* ¼ �<sup>X</sup>

*n*¼1

*ψ*2ð Þ¼ *x*, *λ<sup>m</sup> e*

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

X *N*

*n*¼1

where *cn* ¼ *bn=a*\_ð Þ *λ<sup>n</sup>* . We also define

*λ*2 *m*

*λ*2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � �*λ<sup>n</sup>*

*W*<sup>1</sup> ¼ ð Þ *w*<sup>1</sup> <sup>1</sup>*;*ð Þ *w*<sup>1</sup> <sup>2</sup>*;* ⋯*;*ð Þ *w*<sup>1</sup> *<sup>N</sup>* � �<sup>T</sup>

> *F* ¼ *f* <sup>1</sup>*; f* <sup>2</sup>*;* ⋯*; f <sup>N</sup>* � �<sup>T</sup>

*ψ*1ð Þ¼� *x*, *λ<sup>m</sup>*

*<sup>f</sup> <sup>n</sup>* <sup>¼</sup> ffiffiffiffiffiffi 2*cn* p *e iλ*2 *nx*, *w <sup>j</sup>* � �

(40) can be rewritten as

Then

we know

**29**

ð Þ *B*<sup>1</sup> *mn* ¼ *f <sup>m</sup>*

*n*¼1

*N*

*n*¼1

2*bn a*\_ð Þ *λ<sup>n</sup>*

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

2*bn λna*\_ð Þ *λ<sup>n</sup>*

Let *λ* ¼ *λm*, *m* = 1, 2, … , *N*, respectively, in Eqs. (34) and (35), and make use of

2*λ*<sup>2</sup> *m*

*<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � � *cnψ*1ð Þ *<sup>x</sup>*, *<sup>λ</sup><sup>n</sup> <sup>e</sup>*

> *iλ*2 *nxe* �*iλ*<sup>2</sup>

> > *λm*

, *W*<sup>2</sup> ¼ ð Þ *w*<sup>2</sup> <sup>1</sup>*;*ð Þ *w*<sup>2</sup> <sup>2</sup>*;* ⋯*;*ð Þ *w*<sup>2</sup> *<sup>N</sup>*

*λ* 2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*

, *G* ¼ *f* <sup>1</sup>*=λ*1*; f* <sup>2</sup>*=λ*2*;* ⋯*; f <sup>N</sup>=λ<sup>N</sup>*

where superscript "T" represents transposition of a matrix. Then Eqs. (39) and

*N*

*n*¼1

*N*

*n*¼1

where *m* = 1, 2, … , *N*. They can be rewritten in a more compact matrix form.

*W*<sup>2</sup> ¼ *I* þ *B*1*B*<sup>2</sup>

*W*<sup>1</sup> ¼ �*B*<sup>2</sup> *I* þ *B*1*B*<sup>2</sup>

where *I* is the *N* � *N* identity matrix. On the other hand, from (37) and (38),

� ��<sup>1</sup>

� ��<sup>1</sup>

*λ<sup>n</sup> λ*<sup>2</sup>

*cnψ*2ð Þ *x*, *λ<sup>n</sup> e*

*ψ*2ð Þ *x*, *λ<sup>n</sup> e*

*ψ*1ð Þ *x*, *λ<sup>n</sup> e*

*iλ*2

*iλ*2

<sup>p</sup> *<sup>ψ</sup> <sup>j</sup>*ð Þ *<sup>λ</sup><sup>n</sup> <sup>j</sup>* <sup>¼</sup> 1, 2; and *<sup>n</sup>* <sup>¼</sup> 1, 2, … *<sup>N</sup>:* (41)

� �<sup>T</sup>

� �<sup>T</sup> (43)

ð Þ *B*<sup>1</sup> *mn*ð Þ *w*<sup>1</sup> *<sup>n</sup>* (44)

ð Þ *B*<sup>2</sup> *mn*ð Þ *w*<sup>2</sup> *<sup>n</sup>* (45)

*F* (48)

*F* (49)

*W*<sup>2</sup> ¼ *F* þ *B*<sup>1</sup> � *W*<sup>1</sup> (46) *W*<sup>1</sup> ¼ �*B*<sup>2</sup> � *W*<sup>2</sup> (47)

*iλ*2 *nxe* �*iλ*<sup>2</sup>

*mx*; *<sup>m</sup>* <sup>¼</sup> 1, 2, … , *<sup>N</sup>:* (40)

*f <sup>n</sup>*; *m*, *n* ¼ 1, 2, … , *N* (42)

,

*nx* (37)

*nx* (38)

*mx* (39)

**Figure 1.** *The integral path for IST of the DNLS.*

our modification produces no new poles since Lax operator *L* ! 0, as *λ* ! 0. In the reflectionless case, the revised IST equation gives

$$\Psi\_1(\mathbf{x}, \boldsymbol{\lambda}) = e^{-i\boldsymbol{\lambda}^2 \mathbf{x}} + \sum\_{n=1}^{2N} \frac{\mathbf{1}}{\lambda\_n^2} \frac{\boldsymbol{\lambda}^2}{\boldsymbol{\lambda} - \lambda\_n} \frac{b\_n}{\dot{a}(\lambda\_n)} \boldsymbol{\nu}\_1(\mathbf{x}, \boldsymbol{\lambda}\_n) e^{i\lambda\_n^2 \mathbf{x}} e^{-i\boldsymbol{\lambda}^2 \mathbf{x}} \tag{31}$$

where *a*\_ð Þ¼ *λ<sup>n</sup>* d*a*ð Þ*λ =*d*λ*j *λ*¼*λ<sup>n</sup>* . Similarly, an alternative form of IST equation is proposed as follows:

$$\frac{1}{\lambda} \{\Theta\_2(\varkappa, \lambda)\} e^{i\vec{\lambda}^2 \mathbf{x}} = \frac{1}{2\pi i} \int\_{\Gamma} d\lambda' \frac{1}{\lambda' - \lambda} \frac{1}{\lambda'} \{\Theta\_2(\varkappa, \lambda')\} e^{i\vec{\lambda}'^2 \mathbf{x}} \tag{32}$$

where a factor *λ*�<sup>1</sup> is introduced for the same reason as *λ*�<sup>2</sup> in Eq. (30). Then in the reflectionless case, we can attain

$$\bar{\boldsymbol{\varphi}}\_{2}(\mathbf{x},\boldsymbol{\lambda}) = \sum\_{n=1}^{2N} \frac{1}{\lambda\_{n}} \frac{\boldsymbol{\lambda}}{\lambda - \lambda\_{n}} \frac{b\_{n}}{\dot{\boldsymbol{a}}(\boldsymbol{\lambda}\_{n})} \boldsymbol{\varphi}\_{2}(\mathbf{x},\boldsymbol{\lambda}\_{n}) \boldsymbol{e}^{i\boldsymbol{\lambda}\_{n}^{2}\mathbf{x}} \boldsymbol{e}^{-i\boldsymbol{\lambda}\_{n}^{2}\mathbf{x}} \tag{33}$$

Taking the symmetry and reduction relation (18) and (28) into consideration, from (31) and (33), we can obtain the revised Zakharov-Shabat equation for DNLS equation with VBC, that is,

$$
\tilde{\boldsymbol{\varphi}}\_{1}(\mathbf{x},\boldsymbol{\lambda}) = \boldsymbol{e}^{-i\hat{\boldsymbol{\lambda}}^{2}\mathbf{x}} + \sum\_{n=1}^{N} \frac{2\boldsymbol{\lambda}^{2}}{\lambda\_{n}(\boldsymbol{\lambda}^{2} - \boldsymbol{\lambda}\_{n}^{2})} \frac{\boldsymbol{b}\_{n}}{\dot{\boldsymbol{a}}(\boldsymbol{\lambda}\_{n})} \boldsymbol{\varphi}\_{1}(\mathbf{x}\_{1},\boldsymbol{\lambda}\_{n}) \boldsymbol{e}^{i\hat{\boldsymbol{\lambda}}\_{n}^{2}\mathbf{x}} \boldsymbol{e}^{-i\hat{\boldsymbol{\lambda}}^{2}\mathbf{x}} \tag{34}
$$

$$
\tilde{\boldsymbol{\psi}}\_{2}(\mathbf{x}, \boldsymbol{\lambda}) = \sum\_{n=1}^{N} \frac{2\boldsymbol{\lambda}}{\lambda^{2} - \lambda\_{n}^{2}} \frac{\boldsymbol{b}\_{n}}{\dot{\boldsymbol{a}}(\boldsymbol{\lambda}\_{n})} \boldsymbol{\psi}\_{2}(\mathbf{x}\_{\text{s}}, \boldsymbol{\lambda}\_{n}) e^{i\lambda\_{n}^{2}\mathbf{x}} e^{-i\boldsymbol{\lambda}^{2}\mathbf{x}} \tag{35}
$$

#### **2.2 The raw expression of** *N***-soliton solution**

Substituting Eqs. (34) and (35) into formula (26), we thus attain the *N*-soliton solution

$$
\overline{\mathfrak{u}}\_N = -i2U\_N/V\_N \tag{36}
$$

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

where

$$U\_N = \sum\_{n=1}^N \frac{2b\_n}{\dot{a}(\lambda\_n)} \varphi\_2(\mathbf{x}, \lambda\_n) e^{i\lambda\_n^2 \mathbf{x}} \tag{37}$$

$$V\_N = 1 + \sum\_{n=1}^{N} \frac{2b\_n}{\lambda\_n \dot{a}(\lambda\_n)} \varphi\_1(\mathbf{x}, \lambda\_n) e^{i\lambda\_n^2 \mathbf{x}} \tag{38}$$

Let *λ* ¼ *λm*, *m* = 1, 2, … , *N*, respectively, in Eqs. (34) and (35), and make use of the symmetry and reduction relation (15), we can attain

$$\overline{\mu\_2}(\varkappa;\lambda\_m) = e^{-i\overline{\lambda\_m^2}\varkappa} + \sum\_{n=1}^N \frac{2\overline{\lambda\_m^2}}{\lambda\_n\left(\overline{\lambda\_m^2} - \lambda\_n^2\right)} c\_n \nu\_1(\varkappa;\lambda\_n) e^{i\overline{\lambda\_n^2}\varkappa} e^{-i\overline{\lambda\_m^2}\varkappa} \tag{39}$$

$$\overline{\varphi\_1}(\mathbf{x}, \boldsymbol{\lambda}\_m) = -\sum\_{n=1}^N \frac{2\overline{\lambda\_m^2}}{\overline{\lambda\_m^2} - \overline{\lambda\_n^2}} c\_n \varphi\_2(\mathbf{x}, \boldsymbol{\lambda}\_n) e^{i\overline{\lambda\_n^2} \mathbf{x}} e^{-i\overline{\lambda\_m^2} \mathbf{x}}; \ m = 1, 2, \dots, N. \tag{40}$$

where *cn* ¼ *bn=a*\_ð Þ *λ<sup>n</sup>* . We also define

$$f\_n = \sqrt{2\varepsilon\_n} \varepsilon^{i\lambda\_n^2 \chi}, \ (w\_j)\_n = \sqrt{2\varepsilon\_n} \nu\_j(\lambda\_n) \, j = 1, 2; \text{and } n = 1, 2, \dots \\ N. \tag{41}$$

$$(B\_1)\_{mn} = \overline{f\_m} \frac{\lambda\_m^2}{\left(\overline{\lambda\_m^2} - \lambda\_n^2\right)\lambda\_n} f\_{n^\*} (B\_2)\_{mn} = \overline{f}\_m \frac{\overline{\lambda\_m}}{\overline{\lambda\_m^2} - \lambda\_n^2} f\_n; m, n = 1, 2, \dots, N \tag{42}$$

$$\begin{aligned} W\_1 &= \left( (w\_1)\_1, (w\_1)\_2, \dots, (w\_1)\_N \right)^\mathrm{T}, W\_2 = \left( (w\_2)\_1, (w\_2)\_2, \dots, (w\_2)\_N \right)^\mathrm{T}, \\ F &= \left( f\_1, f\_2, \dots, f\_N \right)^\mathrm{T}, G = \left( f\_1/\lambda\_1 f\_2/\lambda\_2, \dots, f\_N/\lambda\_N \right)^\mathrm{T} \end{aligned} \tag{43}$$

where superscript "T" represents transposition of a matrix. Then Eqs. (39) and (40) can be rewritten as

$$(\overline{w\_2})\_m = \overline{f}\_m + \sum\_{n=1}^{N} (B\_1)\_{mn} (w\_1)\_n \tag{44}$$

$$(\overline{w\_1})\_m = -\sum\_{n=1}^N (B\_2)\_{mn} (w\_2)\_n \tag{45}$$

where *m* = 1, 2, … , *N*. They can be rewritten in a more compact matrix form.

$$
\overline{W}\_2 = \overline{F} + B\_1 \cdot W\_1 \tag{46}
$$

$$
\overline{W}\_1 = -B\_2 \cdot W\_2 \tag{47}
$$

Then

$$W\_2 = \left(I + \overline{B}\_1 B\_2\right)^{-1} F \tag{48}$$

$$\mathcal{W}\_1 = -\overline{B}\_2 \left( I + B\_1 \overline{B}\_2 \right)^{-1} \overline{F} \tag{49}$$

where *I* is the *N* � *N* identity matrix. On the other hand, from (37) and (38), we know

*Nonlinear Optics - From Solitons to Similaritons*

$$U\_N = \sum\_{n=1}^N f\_n w\_{2n} = F^T W\_2 \tag{50}$$

where *Q*<sup>1</sup> *n*1, *n*2, ⋯*nr* ð Þ ; *m*1, *m*2, ⋯, *mr* denotes a minor, which is the determinant of a submatrix of *Q*<sup>1</sup> consisting of elements belonging to not only rows (*n*1, *n*2, … , *nr*) but also columns (*m*1,*m*2, … , *mr*). Here use is made of Binet-Cauchy formula in

*m*, *n*

*<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *m*0 � �<sup>2</sup>

*f <sup>m</sup> f <sup>n</sup> λ* 2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n λm*

*n*, *n*<sup>0</sup> ∈f g *n*1, *n*2, ⋯, *nr* , *m*, *m*<sup>0</sup> ∈f g *m*1, *m*2, ⋯, *mr* (60)

Y *n*< *n*<sup>0</sup> , *m* < *m*<sup>0</sup>

*<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � �<sup>2</sup>

*P*<sup>1</sup> *n*1*;* ⋯*; nr* ð Þ ; *m*1*;* ⋯*; mr P*<sup>2</sup> *m*1*;* ⋯*; mr* ð Þ ; *n*1*;* ⋯*; nr*

,*cn*<sup>0</sup> ¼ *bn*0*=a*\_ð Þ *λ<sup>n</sup>* , *bn*ðÞ¼ *t bn*0*e*

*λ* 2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *m*0 � � *<sup>λ</sup>*<sup>2</sup>

*<sup>n</sup>*<sup>0</sup> � *<sup>λ</sup>*<sup>2</sup> *n* � �

(59)

(61)

(62)

(63)

*i*4*λ*<sup>4</sup> *nt* (64)

*<sup>m</sup>*<sup>0</sup> � *<sup>λ</sup>*<sup>2</sup> *m* � �Y

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

*P*<sup>1</sup> *n*1, *n*2, ⋯, *nr* ð Þ ; *m*1, *m*2, ⋯, *mr P*<sup>2</sup> *m*1, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯, *nr*

Y *n*<*n*<sup>0</sup> , *m* < *m*<sup>0</sup>

*λ*2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *m*0 � �<sup>2</sup> *λ*<sup>2</sup>

It is easy to find a kind of permutation symmetry existed between expressions

Comparing (58) with (62) and making use of (63), we thus complete verifica-

The time evolution factor of the scattering data can be introduced by standard procedure [21]. Due to the fact that the second Lax operator *<sup>M</sup>* ! �*i*2*λ*<sup>4</sup>*σ*<sup>3</sup> in the limit of ∣*x*∣ ! ∞, it is easy to derive the time dependence of scattering date.

> *i*4*λ*<sup>4</sup> *nt*

*P*<sup>1</sup> *n*1, ⋯, *nr* ð Þ ; *m*1, ⋯, *mr P*<sup>2</sup> *m*1, ⋯, *mr* ð Þ ; *n*1, ⋯, *nr* ¼ *Q*<sup>1</sup> *m*1, ⋯, *mr* ð Þ ; *n*1, ⋯, *nr Q*<sup>2</sup> *n*1, ⋯, *nr* ð Þ ; *m*1, ⋯, *mr*

tion of Eq. (55). The soliton solution is surely of a typical form as that in NLS

the Appendices A.2–4 in Part 2. Then

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

Y *n*< *n*<sup>0</sup> , *m* < *m*<sup>0</sup>

<sup>¼</sup> <sup>Q</sup> *n*, *m*

¼ �ð Þ<sup>1</sup> *<sup>r</sup>* <sup>Q</sup> *m*, *n*

where

Similarly,

where

<sup>¼</sup> <sup>1</sup> <sup>þ</sup><sup>X</sup> *N*

**31**

*r*¼1

det *<sup>I</sup>* <sup>þ</sup> *<sup>B</sup>*<sup>1</sup> � *FG*<sup>T</sup> � �*B*<sup>2</sup>

(59) and (61), that is,

*f <sup>n</sup>f <sup>m</sup> λ*2 *<sup>n</sup>* � *λ* 2 *m* *λ*2 *n λm*

*λ*2 *n f* 2 *n f* 2 *m*

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *m* � �<sup>2</sup>

¼ �ð Þ<sup>1</sup> *<sup>r</sup>*

Y *n*, *m* *f* 2 *n f* 2 *mλ*2 *m*

*λ* 2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *m* � �<sup>2</sup>

*n*, *n*<sup>0</sup> ∈f g *n*1*; n*2*;* ⋯*; nr* ; *m*, *m*<sup>0</sup> ∈f g *m*1*; m*2*;* ⋯*; mr* , and

equation and can be expressed as formula (52).

*dλn=dt* ¼ 0, *da*ð Þ *λ<sup>n</sup> =dt* ¼ 0;*cn*ðÞ¼ *t cn*0*e*

*2.3.2 Introduction of time evolution function*

X 1≤ *m*<sup>1</sup> < ⋯ < *mr* ≤ *N*

� � <sup>¼</sup> detð Þ *<sup>I</sup>* <sup>þ</sup> *<sup>P</sup>*1*P*<sup>2</sup>

X 1≤*n*<sup>1</sup> < ⋯ <*nr* ≤ *N*

*Q*<sup>1</sup> *n*1, *n*2, ⋯*nr* ð Þ ; *m*1, *m*2, ⋯, *mr Q*<sup>2</sup> *m*1, *m*2, ⋯*mr* ð Þ ; *n*1, *n*2⋯, *nr*

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � � *λ*<sup>2</sup>

Y *n*<*n*<sup>0</sup> , *m* < *m*<sup>0</sup>

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � �<sup>2</sup> *λ*<sup>2</sup>

$$W\_N = \mathbf{1} + \sum\_{n=1}^{N} \left( f\_n / \lambda\_n \right) w\_{1n} = \mathbf{1} + G^T W\_1 \tag{51}$$

Substituting Eqs. (48), (49) into (50) and (51) and then substituting (50) and (51) into formula (36), we thus attain

$$\begin{split} \overline{u}\_{N} &= -i2 \frac{F^{\mathrm{T}} W\_{2}}{1 + G^{\mathrm{T}} W\_{1}} = -i2 \frac{F^{\mathrm{T}} \left(I + \overline{B}\_{1} B\_{2}\right)^{-1} \overline{F}}{1 - G^{\mathrm{T}} \overline{B}\_{2} \left(I + B\_{1} \overline{B}\_{2}\right)^{-1} \overline{F}} \\ &= -i2 \frac{\det\left(I + \overline{B}\_{1} B\_{2} + F F^{\mathrm{T}}\right) - \det\left(I + \overline{B}\_{1} B\_{2}\right)}{\det\left[I + \left(B\_{1} - \overline{F} G^{\mathrm{T}}\right) \overline{B}\_{2}\right]} \cdot \frac{\det\left(I + B\_{1} \overline{B}\_{2}\right)}{\det\left(I + \overline{B}\_{1} B\_{2}\right)} \equiv -2i \frac{A \cdot D}{\overline{D}^{2}} \end{split} \tag{52}$$

where

$$A \equiv \det\left(I + \overline{B}\_1 B\_2 + F F^T\right) - \det\left(I + \overline{B}\_1 B\_2\right) \tag{53}$$

$$D \equiv \det(I + B\_1 \overline{B}\_2) \tag{54}$$

In the subsequent chapter, we will prove that

$$\det\left[I + \left(B\_1 - \overline{F}G^T\right)\overline{B}\_2\right] = \det(I + \overline{B}\_1 B\_2) \tag{55}$$

It is obvious that formula (52) has the usual standard form of soliton solution. Here in formula (52), some algebra techniques have been used and can be found in Appendix A.1 in Part 2.

#### **2.3 Explicit expression of** *N***-soliton solution**

#### *2.3.1 Verification of standard form for the N-soliton solution*

We only need to prove that Eq. (55) holds. Firstly, we define *N* � *N* matrices *P*1, *P*2, *Q*1, *Q*2, respectively, as

$$\left( (P\_1)\_{nm} \equiv \left( B\_1 - \overline{F} G^T \right)\_{nm} = \overline{f}\_n \frac{\lambda\_m}{\overline{\lambda\_n^2} - \lambda\_m^2} f\_m; \left( P\_2 \right)\_{mn} \equiv \left( \overline{B}\_2 \right)\_{mn} = f\_m \frac{\lambda\_m}{\lambda\_m^2 - \overline{\lambda\_n^2}} \overline{f}\_n \tag{56}$$

$$(\left(Q\_1\right)\_{nm}\equiv \left(\overline{B}\_1\right)\_{nm} = f\_n \frac{\lambda\_n^2}{\lambda\_n^2 - \overline{\lambda\_m^2}} \left(\frac{\overline{f}\_m}{\overline{\lambda}\_m}\right); \left(Q\_2\right)\_{mn} \equiv \left(B\_2\right)\_{mn} = \overline{f}\_m \frac{\overline{\lambda}\_m}{\overline{\lambda\_m^2} - \lambda\_n^2} f\_n \tag{57}$$

Then

$$\overline{D} = \det(I + Q\_1 Q\_2) = \mathbf{1} + \sum\_{r=1}^{N} \sum\_{1 \le n\_1 < n\_2 < \cdots < n\_r \le N} \overline{D}\_r(n\_1, n\_2, \cdots, n\_r)$$

$$= \mathbf{1} + \sum\_{r=1}^{N} \sum\_{1 \le n\_1 < \cdots < n\_r \le N} \sum\_{1 \le m\_1 < \cdots < m\_r \le N} Q\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r) Q\_2(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r) \tag{58}$$

#### *Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

where *Q*<sup>1</sup> *n*1, *n*2, ⋯*nr* ð Þ ; *m*1, *m*2, ⋯, *mr* denotes a minor, which is the determinant of a submatrix of *Q*<sup>1</sup> consisting of elements belonging to not only rows (*n*1, *n*2, … , *nr*) but also columns (*m*1,*m*2, … , *mr*). Here use is made of Binet-Cauchy formula in the Appendices A.2–4 in Part 2. Then

$$Q\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r) Q\_2(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r)$$

$$=\prod\_{n,m} \frac{f\,\overline{f}\_{m}}{\lambda\_{n}^{2}-\overline{\lambda}\_{m}^{2}}\frac{\lambda\_{n}^{2}}{\overline{\lambda}\_{m}}\prod\_{n
$$=(-1)^{r}\prod\_{m,n}\frac{\lambda\_{n}^{2}f^{2}\overline{f\_{m}^{2}}}{\left(\lambda\_{n}^{2}-\overline{\lambda}\_{m}^{2}\right)^{2}}\prod\_{n$$
$$

where

$$\{n, n' \in \{n\_1, n\_2, \dots, n\_r\}, m, m' \in \{m\_1, m\_2, \dots, m\_r\}\tag{60}$$

Similarly,

$$\begin{split} &P\_1(n\_1, n\_2, \ldots, n\_r; m\_1, m\_2, \ldots, m\_r) P\_2(m\_1, m\_2, \ldots, m\_r; n\_1, n\_2, \ldots, n\_r) \\ &= (-1)^r \prod\_{n,m} \frac{\overline{\widehat{\lambda}\_n^2} f\_m^2 \lambda\_m^2}{\left(\overline{\lambda}\_n^2 - \lambda\_m^2\right)^2} \prod\_{n < n', m < m'} \left(\lambda\_m^2 - \lambda\_{m'}^2\right)^2 \left(\overline{\lambda\_n^2} - \overline{\lambda\_{n'}^2}\right)^2 \end{split} \tag{61}$$

where

$$\begin{aligned} &m, n' \in \{n\_1, n\_2, \dots, n\_r\}; m, m' \in \{m\_1, m\_2, \dots, m\_r\}, \text{and} \\ &\det[I + (B\_1 - \overline{F}G^\mathrm{T})\overline{B}\_2] = \det(I + P\_1 P\_2) \\ &= 1 + \sum\_{r=1}^N \sum\_{1 \le n\_1 < \dots < n\_r \le N} \sum\_{1 \le m\_1 < \dots < m\_r \le N} P\_1(n\_1, \dots, n\_r; m\_1, \dots, m\_r) P\_2(m\_1, \dots, m\_r; n\_1, \dots, n\_r) \end{aligned} \tag{62}$$

It is easy to find a kind of permutation symmetry existed between expressions (59) and (61), that is,

$$\begin{aligned} P\_1(n\_1, \ldots, n\_r; m\_1, \ldots, m\_r) P\_2(m\_1, \ldots, m\_r; n\_1, \ldots, n\_r) \\ = Q\_1(m\_1, \ldots, m\_r; n\_1, \ldots, n\_r) Q\_2(n\_1, \ldots, n\_r; m\_1, \ldots, m\_r) \end{aligned} \tag{63}$$

Comparing (58) with (62) and making use of (63), we thus complete verification of Eq. (55). The soliton solution is surely of a typical form as that in NLS equation and can be expressed as formula (52).

#### *2.3.2 Introduction of time evolution function*

The time evolution factor of the scattering data can be introduced by standard procedure [21]. Due to the fact that the second Lax operator *<sup>M</sup>* ! �*i*2*λ*<sup>4</sup>*σ*<sup>3</sup> in the limit of ∣*x*∣ ! ∞, it is easy to derive the time dependence of scattering date.

$$d\lambda\_n/dt = 0, da(\lambda\_n)/dt = 0; c\_n(t) = c\_{n0}e^{i4\lambda\_n^4t}, c\_{n0} = b\_{n0}/\dot{a}(\lambda\_n), b\_n(t) = b\_{n0}e^{i4\lambda\_n^4t} \tag{64}$$

Then the typical soliton arguments *θ<sup>n</sup>* and *φ<sup>n</sup>* can be defined according to

$$f\_n^2 = 2c\_{n0}e^{i2\dot{\lambda}\_n^2 \chi} e^{i4\dot{\lambda}\_n^4 t} \equiv 2c\_{n0}e^{-\theta\_n}e^{i\rho\_n} \tag{65}$$

Ω<sup>1</sup> *n*1, *n*2, ⋯, *nr* ð Þ ; 0, *m*2, ⋯, *mr* Ω<sup>2</sup> 0, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯, *nr*

Y *n* <*n*<sup>0</sup> , *m* < *m*<sup>0</sup>

**2.4 The typical examples for one- and two-soliton solutions**

*D*<sup>1</sup> ¼ *Q*1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 1 *Q*2ð*m*<sup>1</sup> ¼ 1; *n*<sup>2</sup> ¼ 1Þ ¼ 1 � *f* <sup>1</sup>

;*c*<sup>10</sup> ¼ *b*10*=a*\_ð Þ *λ*<sup>1</sup> ; *b*<sup>10</sup> ¼ *e*

<sup>1</sup> � *<sup>ν</sup>*<sup>2</sup> 1 � �*<sup>t</sup>* � �; *<sup>φ</sup>*<sup>1</sup> <sup>¼</sup> <sup>2</sup> *<sup>μ</sup>*<sup>2</sup>

absorbed into the soliton center and initial phase. Then

<sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �*<sup>e</sup>*

> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1

> > j j *λ*1 2

� � �

*<sup>A</sup>*<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>1</sup> *<sup>λ</sup>*<sup>2</sup>

*D*<sup>1</sup> ¼ 1 �

*<sup>u</sup>*1ð Þ¼� *<sup>x</sup>*, *<sup>t</sup> <sup>i</sup>*2*A*1*D*1*=D*<sup>2</sup>

tions of the general explicit soliton solution.

�*θ<sup>n</sup> e <sup>i</sup>φ<sup>n</sup> e* �*θ<sup>m</sup> e*

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � �<sup>2</sup> *λ*<sup>2</sup>

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

�*iφ<sup>m</sup> <sup>λ</sup>*<sup>2</sup>

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *m* � �<sup>2</sup>

Here *n*, *n*<sup>0</sup> ∈f g *n*1, *n*2, ⋯, *nr* and especially *m*, *m*<sup>0</sup> ∈f g *m*2, ⋯, *mr* , which completes the calculation of determinant *A* in formula (52). Substituting the explicit expressions of *D*, *D*, and *A* into (52), we finally attain the explicit expression of *N*-soliton solution to the DNLS equation under VBC and reflectionless case, based upon a

An interesting conclusion is found that, besides a permitted well-known constant global phase factor, there is also an undetermined constant complex parameter *bn*<sup>0</sup> before each of the typical soliton factor *e*�*θ<sup>n</sup> e<sup>i</sup>φ<sup>n</sup>* , (*n* = 1,2, … , *N*). It can be absorbed into *e*�*θ<sup>n</sup> e<sup>i</sup>φ<sup>n</sup>* by redefinition of soliton center and its initial phase factor. This kind of arbitrariness is in correspondence with the unfixed initial conditions of

We give two concrete examples – the one- and two-soliton solutions as illustra-

In the case of one-soliton solution, *<sup>N</sup>* = 1, *<sup>λ</sup>*<sup>2</sup> ¼ �*λ*1, *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>*1*e<sup>i</sup>β*<sup>1</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>i</sup>ν*1, and

*A*<sup>1</sup> ¼ Ω1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 0 Ω2ð*m*<sup>1</sup> ¼ 0; *n*<sup>1</sup> ¼ 1Þ ¼ *f*

It is different slightly from the definition in (66) for that here *b*<sup>10</sup> has been

*λ*2 1

<sup>1</sup> <sup>¼</sup> <sup>4</sup>*ρ*<sup>1</sup> sin 2*β*1*e<sup>i</sup>*3*β*<sup>1</sup> <sup>1</sup> <sup>þ</sup> *<sup>e</sup>*�*i*2*β*<sup>1</sup> *<sup>e</sup>*�2*θ*<sup>1</sup>

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *<sup>e</sup>*

�*θ*<sup>1</sup> *e <sup>i</sup>φ*1*=λ* 2

� � � 2 <sup>4</sup>*μ*1*ν*1*x*<sup>10</sup> *e*

<sup>1</sup> � *<sup>ν</sup>*<sup>2</sup> 1 � �*<sup>x</sup>* <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

<sup>1</sup> ¼ *i*2*ρ*<sup>1</sup> sin 2*β*1*e*

�2*θ*<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>e</sup>*

� � <sup>1</sup> <sup>þ</sup> *ei*2*β*<sup>1</sup> *<sup>e</sup>* ð Þ �2*θ*<sup>1</sup> <sup>2</sup> *<sup>e</sup>*

*m*

*<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *m*0 � �<sup>2</sup>

> Y *n*<*n*<sup>0</sup> , *m* < *m*<sup>0</sup>

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � �<sup>2</sup> *λ*<sup>2</sup>

2

� � � � 4 *λ*2 <sup>1</sup>*= λ*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup>

> *i*2*λ*<sup>2</sup> 1*xe i*4*λ*<sup>4</sup> <sup>1</sup> *<sup>t</sup>* � *<sup>e</sup>* �*θ*<sup>1</sup> *e <sup>i</sup>φ*<sup>1</sup> ;

<sup>1</sup> � *<sup>ν</sup>*<sup>2</sup> 1 � �<sup>2</sup> � <sup>16</sup>*μ*1*ν*<sup>2</sup>

*<sup>i</sup>*3*β*<sup>1</sup> *e* �*θ*<sup>1</sup> *e iφ*<sup>1</sup>

> *<sup>i</sup>*2*β*<sup>1</sup> *e* �2*θ*<sup>1</sup>

> > �*θ*<sup>1</sup> *e*

*<sup>i</sup>φ*<sup>1</sup> (75)

*<sup>i</sup>α*<sup>10</sup> ; *b*10*e*

<sup>1</sup> (72)

1

h i*<sup>t</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>10</sup>

(73)

(74)

*<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *m*0 � �<sup>2</sup>

(71)

¼ �ð Þ<sup>1</sup> *<sup>r</sup>*�<sup>1</sup>

¼ �ð Þ<sup>1</sup> *<sup>r</sup>*�<sup>1</sup>

Y *n*, *m*

Y *n*, *m*

newly revised IST technique.

the DNLS equation.

*i*2*λ*<sup>2</sup> 1*xe i*4*λ*<sup>4</sup> 1 *t*

*<sup>θ</sup>*<sup>1</sup> <sup>¼</sup> <sup>4</sup>*μ*1*ν*<sup>1</sup> *<sup>x</sup>* � *<sup>x</sup>*<sup>10</sup> <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

*f* 2 <sup>1</sup> ¼ 2*c*10*e*

and

**33**

*f* 2 *n f* 2 *mλ*2 *m*

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *m* � �<sup>2</sup>

ð Þ 2*cn* ð Þ 2*cm e*

where *<sup>λ</sup>* � *<sup>μ</sup><sup>n</sup>* <sup>þ</sup> *<sup>i</sup>νn*, and *<sup>θ</sup><sup>n</sup>* <sup>¼</sup> <sup>4</sup>*μnν<sup>n</sup> <sup>x</sup>* <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>n</sup>* � *<sup>ν</sup>*<sup>2</sup> *n* � �*<sup>t</sup>* � � <sup>¼</sup> <sup>4</sup>*κn*ð Þ *<sup>x</sup>* � *Vnt* ; *<sup>φ</sup><sup>n</sup>* <sup>¼</sup> <sup>2</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>n</sup>* � *<sup>ν</sup>*<sup>2</sup> *n* � �*<sup>x</sup>* <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>n</sup>* � *<sup>ν</sup>*<sup>2</sup> *n* � �<sup>2</sup> � <sup>16</sup>*μ*<sup>2</sup> *nν*2 *n* h i � *<sup>t</sup>*

$$V\_n = -4\left(\mu\_n^2 - \nu\_n^2\right), \kappa\_n = 4\mu\_n\nu\_n\tag{66}$$

#### *2.3.3 Calculation of determinant of D and A*

Substituting expression (64) and (65) into formula (59) and then into (58), we have

$$Q\_1(n\_1, n\_2, \dots, n\_r; m\_1, m\_2, \dots, m\_r) Q\_2(m\_1, m\_2, \dots, m\_r; n\_1, n\_2, \dots, n\_r)$$

$$Q\_r = (-1)^r \prod\_{n\_r m} (2c\_n) (2\overline{c}\_m) e^{-\theta\_n} e^{i\rho\_n} e^{-\theta\_n} e^{-i\rho\_n} \frac{\lambda\_n^2}{\left(\lambda\_n^2 - \overline{\lambda\_m^2}\right)^2} \prod\_{n < n', m < m'} \left(\lambda\_n^2 - \lambda\_{n'}^2\right)^2 \left(\overline{\lambda\_m^2} - \overline{\lambda\_{m'}^2}\right)^2 \tag{57}$$

with *n*, *n*<sup>0</sup> ∈f g *n*1, *n*2, ⋯, *nr* and *m*, *m*<sup>0</sup> ∈f g *m*1, *m*2, ⋯, *mr* . Where use is made of Binet-Cauchy formula which is numerated in Appendix A. 3–4 in Part 2. Substituting expression (67) into formula (58) thus complete the calculation of determinant *D*.

About the calculation of the most complicate determinant *A* in (52), we introduce a *N*�(*N* + 1) matrix Ω<sup>1</sup> and a (*N* + 1) � *N* matrix Ω<sup>2</sup> defined as

$$\begin{aligned} \left(\Omega\_1\right)\_{nm} &= \left(\overline{B}\_1\right)\_{nm} = \left(Q\_1\right)\_{nm}, \left(\Omega\_1\right)\_{n0} = f\_n, \\ \left(\Omega\_2\right)\_{mn} &= \left(B\_2\right)\_{mn} = \left(Q\_2\right)\_{mn}, \left(\Omega\_2\right)\_{0n} = f\_n \end{aligned} \tag{68}$$

with *n*, *m* ¼ 1, 2, ⋯, *N*. We thus have

$$\begin{split} \det(I + \overline{B}\_1 B\_2 + F F^T) &= \det(I + \mathfrak{Q}\_1 \Omega\_2) \\ = 1 + \sum\_{r=1}^N \sum\_{1 \le n\_1 < \cdots < n\_r \le N} \sum\_{0 \le m\_1 < \cdots < m\_r \le N} \Omega\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r) \Omega\_2(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r) \end{split} \tag{69}$$

The above summation obviously can be decomposed into two parts: one is extended to *m*<sup>1</sup> = 0, the other is extended to *m*<sup>1</sup> ≥1. Subtracted from (69), the part that is extended to *m*<sup>1</sup> ≥ 1, the remaining parts of (69) is just *A* in (52) (with *m*<sup>1</sup> ¼ 0 and *m*<sup>2</sup> ≥1). Due to (68), we thus have

$$\begin{aligned} A &= \det(I + \Omega\_1 \Omega\_2) - \det(I + Q\_1 Q\_2) \\ &= \sum\_{r=1}^N \sum\_{1 \le n\_1 < \cdots < n\_r \le N} \sum\_{1 \le m\_1 < \cdots < m\_r \le N} A\_r(n\_1, n\_2, \cdots, n\_r; 0, m\_2, \cdots, m\_r) \\ &= \sum\_{r=1}^N \sum\_{1 \le n\_1 < n\_2 < n\_r} \sum\_{1 \le m\_2 < m\_1 < \cdots < m\_r \le N} \Omega\_1(n\_1, n\_2, \cdots, n\_r; 0, m\_2, \cdots, m\_r) \Omega\_2(0, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r) \end{aligned} \tag{70}$$

with

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

Ω<sup>1</sup> *n*1, *n*2, ⋯, *nr* ð Þ ; 0, *m*2, ⋯, *mr* Ω<sup>2</sup> 0, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯, *nr*

$$=(-1)^{r-1} \prod\_{n,m} \frac{f\_n^2 f\_m^2 \lambda\_m^2}{\left(\lambda\_n^2 - \overline{\lambda\_m^2}\right)^2} \prod\_{n
$$= (-1)^{r-1} \prod\_{n,m} (2c\_n)(2\overline{c\_m})e^{-\theta\_n}e^{i\varphi\_n}e^{-\theta\_n}e^{-i\varphi\_n} \frac{\overline{\lambda\_m^2}}{\left(\lambda\_n^2 - \overline{\lambda\_m^2}\right)^2} \prod\_{n$$
$$

Here *n*, *n*<sup>0</sup> ∈f g *n*1, *n*2, ⋯, *nr* and especially *m*, *m*<sup>0</sup> ∈f g *m*2, ⋯, *mr* , which completes the calculation of determinant *A* in formula (52). Substituting the explicit expressions of *D*, *D*, and *A* into (52), we finally attain the explicit expression of *N*-soliton solution to the DNLS equation under VBC and reflectionless case, based upon a newly revised IST technique.

An interesting conclusion is found that, besides a permitted well-known constant global phase factor, there is also an undetermined constant complex parameter *bn*<sup>0</sup> before each of the typical soliton factor *e*�*θ<sup>n</sup> e<sup>i</sup>φ<sup>n</sup>* , (*n* = 1,2, … , *N*). It can be absorbed into *e*�*θ<sup>n</sup> e<sup>i</sup>φ<sup>n</sup>* by redefinition of soliton center and its initial phase factor. This kind of arbitrariness is in correspondence with the unfixed initial conditions of the DNLS equation.

#### **2.4 The typical examples for one- and two-soliton solutions**

We give two concrete examples – the one- and two-soliton solutions as illustrations of the general explicit soliton solution.

In the case of one-soliton solution, *<sup>N</sup>* = 1, *<sup>λ</sup>*<sup>2</sup> ¼ �*λ*1, *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>*1*e<sup>i</sup>β*<sup>1</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>i</sup>ν*1, and

$$A\_1 = \Omega\_1(n\_1 = 1; m\_1 = 0)\Omega\_2(m\_1 = 0; n\_1 = 1) = f\_1^2 \tag{72}$$

$$\overline{D}\_1 = Q\_1(n\_1 = 1; m\_1 = 1) Q\_2(m\_1 = 1; n\_2 = 1) = 1 - \left| f\_1 \right|^4 \lambda\_1^2 / \left( \lambda\_1^2 - \overline{\lambda\_1^2} \right)^2 \tag{73}$$

$$f\_1^2 = 2c\_{10}e^{i2\hat{\ell}\_1^x \mathbf{x}}e^{i4\hat{\ell}\_1^t t}; c\_{10} = b\_{10}/\dot{a}\,(\lambda\_1); b\_{10} = e^{4\mu\_1\mu\_{10}}e^{i\alpha\_0}; b\_{10}e^{i2\hat{\ell}\_1^x \mathbf{x}}e^{i4\hat{\ell}\_1^t t} \equiv e^{-\theta\_1}e^{i\rho\_1};$$

$$\theta\_1 = 4\mu\_1\nu\_1 \left[\mathbf{x} - \mathbf{x}\_{10} + 4\left(\mu\_1^2 - \nu\_1^2\right)t\right]; \rho\_1 = 2\left(\mu\_1^2 - \nu\_1^2\right)\mathbf{x} + \left[4\left(\mu\_1^2 - \nu\_1^2\right)^2 - 16\mu\_1\nu\_1^2\right]t + a\_{10} \tag{74}$$

It is different slightly from the definition in (66) for that here *b*<sup>10</sup> has been absorbed into the soliton center and initial phase. Then

$$A\_1 = \lambda\_1 \left(\lambda\_1^2 - \overline{\lambda\_1^2}\right) e^{-\theta\_1} e^{i\varphi\_1} / \overline{\lambda\_1^2} = i2\rho\_1 \sin 2\beta\_1 e^{i3\beta\_1} e^{-\theta\_1} e^{i\varphi\_1}$$

$$\overline{D}\_1 = \mathbf{1} - \frac{\left|\lambda\_1^2 - \overline{\lambda\_1^2}\right|^2}{\left|\lambda\_1\right|^2} \frac{\lambda\_1^2}{\left(\lambda\_1^2 - \overline{\lambda\_1^2}\right)^2} e^{-2\theta\_1} = \mathbf{1} + e^{i2\beta\_1} e^{-2\theta\_1}$$

and

$$
\overline{u}\_1(\mathbf{x}, t) = -i2A\_1 D\_1 / \overline{D}\_1^2 = \frac{4\rho\_1 \sin 2\beta\_1 e^{i3\beta\_1} \left(\mathbf{1} + e^{-i2\beta\_1} e^{-2\theta\_1}\right)}{\left(\mathbf{1} + e^{i2\beta\_1} e^{-2\theta\_1}\right)^2} e^{-\theta\_1} e^{i\phi\_1} \tag{75}
$$

The complex conjugate of one-soliton solution *u*1ð Þ *x*, *t* in (75) is *u*1ð Þ *x*, *t* , which is just in conformity with that gotten from pure Marchenko formalism [24] (see the next section), up to a permitted global constant phase factor. In the case of twosoliton solution, *N* = 2, *λ*<sup>3</sup> ¼ �*λ*1, *λ*<sup>4</sup> ¼ �*λ*<sup>2</sup> and

$$
\lambda\_1 = \rho\_1 e^{i\theta\_1} = \mu\_1 + i\nu\_1,\\
\lambda\_2 = \rho\_2 e^{i\theta\_2} = \mu\_2 + i\nu\_2 \tag{76}
$$

$$\begin{aligned} c\_{10} &= \frac{b\_{10}}{\dot{a}(\lambda\_1)} = b\_{10} \frac{\lambda\_1^2 - \lambda\_1^2}{2\lambda\_1} \cdot \frac{\lambda\_1^2 - \lambda\_2^2}{\lambda\_1^2 - \lambda\_2^2} \cdot \frac{\lambda\_1^2}{\overline{\lambda\_1^2}} \cdot \frac{\lambda\_2^2}{\overline{\lambda\_2^2}} \\ c\_{20} &= \frac{b\_{20}}{\dot{a}(\lambda\_2)} = b\_{20} \frac{\lambda\_2^2 - \overline{\lambda\_2}}{2\lambda\_2} \cdot \frac{\lambda\_2^2 - \overline{\lambda\_1^2}}{\lambda\_2^2 - \lambda\_1^2} \cdot \frac{\lambda\_1^2}{\overline{\lambda\_1^2}} \cdot \frac{\lambda\_2^2}{\overline{\lambda\_2^2}} \end{aligned} \tag{77}$$

1

2

*<sup>D</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>X<sup>2</sup>

¼ 1 � *f* <sup>1</sup> � � � � <sup>4</sup> *λ*<sup>2</sup> 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> � *<sup>f</sup>* <sup>2</sup>

þ *f* <sup>1</sup>*f* <sup>2</sup> � � � � 4 � *r*¼1

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup>

> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2

� � � � �

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2

<sup>þ</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup>

@

þ *e*

Liouville theorem [25].

*N*-soliton solution as

**35**

*D*<sup>2</sup> ¼ 1 þ

where

X 1 ≤*n*<sup>1</sup> <*n*<sup>2</sup> ≤2

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

*λ*2 1*λ*2 <sup>2</sup> *λ*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup>

> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup>

> > � � � � �

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2

<sup>2</sup> 0

� � � � �

�*i*<sup>2</sup> *<sup>β</sup>*1þ*β*<sup>2</sup> ð Þ*e*

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2

� � � � �

tion to the DNLS equation with VBC

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2

2 *e* �*i*2*β*<sup>1</sup> *e*

<sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � � � � �

�2ð Þ *θ*1þ*θ*<sup>2</sup>

� � � � �

**2.5 The asymptotic behaviors of** *N***-soliton solution**

as Γ*<sup>n</sup>* : *x* � *xno* � *Vnt* � 0,ð Þ *n* ¼ 1*:*2, ⋯, *N* .

2

1 A*e*

X 1 ≤ *m*<sup>1</sup> < *m*<sup>2</sup> ≤2

> � � � � <sup>4</sup> *λ*<sup>2</sup> 2 *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *<sup>f</sup>*

¼ 1 þ *Q*1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 1 *Q*2ð*m*<sup>1</sup> ¼ 1*; n*<sup>1</sup> ¼ 1Þ þ *Q*1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 2*; n*<sup>1</sup> ¼ 1 þ*Q*1ð Þ *n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 1 *Q*2ð*m*<sup>1</sup> ¼ 1*; n*<sup>1</sup> ¼ 2Þ þ *Q*1ð Þ *n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 2*; n*<sup>1</sup> ¼ 2

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

*λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup>

�*i*2*β*<sup>2</sup> *e* �2*θ*<sup>2</sup>

�*<sup>i</sup> <sup>β</sup>*1þ*β*<sup>2</sup> ð Þ *<sup>ρ</sup>*<sup>1</sup>

<sup>¼</sup> *<sup>ρ</sup>*1*=ρ*<sup>2</sup> � *<sup>ρ</sup>*2*=ρ*<sup>1</sup> ð Þ<sup>2</sup> <sup>þ</sup> 4 sin <sup>2</sup> *<sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> ð Þ

Substituting (81) and (84) into formula (52), we thus get the two-soliton solu-

*<sup>u</sup>*<sup>2</sup> ¼ �*i*2*A*2*D*2*=D*<sup>2</sup>

Once again we find that, up to a permitted global constant phase factor, the above two-soliton solution is equivalent to that gotten in Ref. [23, 24], verifying the validity of our formula of *N*-soliton solution and the reliability of those linear algebra techniques. As a matter of fact, a general and strict demonstration of our revised IST for DNLS equation with VBC has been given in one paper by use of

The complex conjugate of expression (52) gives the explicit expression of

*uN* <sup>¼</sup> *<sup>i</sup>*2*ANDN=D*<sup>2</sup>

*n* ¼ 1, 2, ⋯, *N*, we assume *V*<sup>1</sup> <*V*<sup>2</sup> < ⋯ <*Vn* < ⋯*VN* and define the *n*'th vicinity area

As *t* ! �∞, *N* vicinity areas Γ*n*, *n* ¼ 1, 2… , *N*, queue up in a descending series

Without the loss of generality**,** for *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *<sup>μ</sup><sup>n</sup>* <sup>þ</sup> *ivn*,*V*<sup>n</sup> ¼ �<sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

and in the vicinity of Γ*n*, we have (note that *κ <sup>j</sup>* >0)

*ρ*2 *e*

*λ*2 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *<sup>f</sup>*

þ*Q*1ð Þ *n*<sup>1</sup> ¼ 1*; n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 1*; m*<sup>2</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 1*; m*<sup>2</sup> ¼ 2; *n*<sup>1</sup> ¼ 1*; n*<sup>2</sup> ¼ 2

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup>

�2*θ*<sup>1</sup> <sup>þ</sup> *<sup>e</sup>*

� �

�ð Þ *<sup>θ</sup>*1þ*θ*<sup>2</sup> *e*

*λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup>

*Q*<sup>1</sup> *n*1*;* ⋯*; nr* ð Þ ; *m*1*;* ⋯*; mr Q*<sup>2</sup> *m*1*;* ⋯*; mr* ð Þ ; *n*1*;* ⋯*; nr*

*<sup>i</sup> <sup>φ</sup>*2�*φ*<sup>1</sup> ð Þ <sup>þ</sup> *<sup>ρ</sup>*<sup>2</sup>

*<sup>ρ</sup>*1*=ρ*<sup>2</sup> � *<sup>ρ</sup>*2*=ρ*<sup>1</sup> ð Þ<sup>2</sup> <sup>þ</sup> 4 sin <sup>2</sup> *<sup>β</sup>*<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> ð Þ (85)

*ρ*1 *e <sup>i</sup> <sup>φ</sup>*1�*φ*<sup>2</sup> ð Þ � �

<sup>2</sup> (86)

*<sup>N</sup>* (87)

*<sup>n</sup>* � *<sup>v</sup>*<sup>2</sup> *n* � �,

Γ*N*, Γ*<sup>N</sup>*�1, ⋯, Γ<sup>1</sup> (88)

*λ*2 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> (83)

(84)

$$\mathbf{1}\_{j}\mathbf{1}\_{j}^{2} = 2\mathbf{c}\_{j0}e^{j2\mathbf{l}\_{j}^{2}\mathbf{x}}e^{i4\mathbf{l}\_{j}^{4}\mathbf{t}}, j = \mathbf{1}, \mathbf{2}\left(\mathbf{C}f.(\mathbf{1}.62)\right), \\
\mathbf{b}\_{j0}e^{j2\mathbf{l}\_{j}^{2}\mathbf{x} + i4\mathbf{l}\_{j}^{4}\mathbf{t}} \equiv e^{-\theta\_{j}}e^{i\boldsymbol{p}\_{j}}, j = \mathbf{1}, \mathbf{2} \tag{78}$$

where *<sup>θ</sup> <sup>j</sup>* <sup>¼</sup> <sup>4</sup>*<sup>μ</sup> <sup>j</sup><sup>ν</sup> <sup>j</sup> <sup>x</sup>* � *<sup>x</sup> <sup>j</sup>*<sup>0</sup> <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>j</sup>* � *<sup>ν</sup>*<sup>2</sup> *j* � �*<sup>t</sup>* h i

$$\boldsymbol{\rho}\_{j} = 2\left(\boldsymbol{\mu}\_{j}^{2} - \boldsymbol{\nu}\_{j}^{2}\right)\mathbf{x} + \left[4\left(\boldsymbol{\mu}\_{j}^{2} - \boldsymbol{\nu}\_{j}^{2}\right)^{2} - 16\boldsymbol{\mu}\_{j}^{2}\boldsymbol{\nu}\_{j}^{2}\right] \cdot \mathbf{t} + \boldsymbol{a}\_{j0} \tag{79}$$

and *b <sup>j</sup>*<sup>0</sup> is absorbed into the soliton center and the initial phase by

$$b\_{~j0} = e^{4\mu\_{~j}\nu\_{~j0}}e^{ia\_{~j0}}; j=1,2\tag{80}$$

And we get

$$\begin{aligned} A\_2 &= \sum\_{\substack{n\_1=1,2\\n\_1=1,\ldots,n\_2=1}} \Omega\_1(n\_1,0)\Omega\_2(0,n\_1) + \sum\_{\substack{n\_1=1,n\_2=2\\n\_1=0,\ldots,n\_2=1,2}} \Omega\_1(n\_1,n\_2;0,m\_2)\Omega\_2(0,m\_2;m\_1,n\_2) \\ m\_1 &= 0, m\_2 = 1,2 \end{aligned}$$

¼ Ω1ð Þ 1; 0 Ω2ð Þþ 0; 1 Ω1ð Þ 2; 0 Ω2ð Þþ 0; 2 Ω1ð Þ 1*;* 2; 0*;* 1 Ω2ð Þþ 0*;* 1; 1*;* 2 Ω1ð Þ 1*;* 2; 0*;* 2 Ω2ð Þ 0*;* 2; 1*;* 2

¼ *f* 2 <sup>1</sup> þ *f* 2 <sup>2</sup> � *f* <sup>1</sup> � � � � 4 *f* 2 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *<sup>f</sup>* <sup>2</sup> � � � �4 *f* 2 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 2 *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>1</sup> <sup>1</sup> � *<sup>e</sup>*�*i*4*β*<sup>1</sup> � � *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *e <sup>i</sup>*<sup>4</sup> *<sup>β</sup>*1þ*β*<sup>2</sup> ð Þ*e* �*θ*<sup>1</sup> *e <sup>i</sup>φ*<sup>1</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>e</sup>* �*i*4*β*<sup>2</sup> � � *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *e <sup>i</sup>*<sup>4</sup> *<sup>β</sup>*1þ*β*<sup>2</sup> ð Þ*e* �*θ*<sup>2</sup> *e iφ*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>1</sup> <sup>1</sup> � *<sup>e</sup>*�*i*4*β*<sup>1</sup> � �*e*�*i*2*β*<sup>2</sup> *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *e* �*θ*2�*iφ*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>e</sup>* �*i*4*β*<sup>2</sup> � �*e* �*i*2*β*<sup>1</sup> *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *e* �*θ*1�*iφ*<sup>1</sup> " #•*e*�ð Þ *<sup>θ</sup>*1þ*θ*<sup>2</sup> *e<sup>i</sup> <sup>φ</sup>*1þ*φ*<sup>2</sup> ð Þ*ei*<sup>4</sup> *<sup>β</sup>*1þ*β*<sup>2</sup> ð Þ <sup>¼</sup> *<sup>i</sup>*<sup>2</sup> *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � � � � � � � � � � <sup>½</sup>*ρ*<sup>1</sup> sin 2*β*1*ei*ð Þ *<sup>φ</sup>*�*<sup>α</sup> <sup>e</sup><sup>i</sup>* <sup>3</sup>*β*1þ4*β*<sup>2</sup> ð Þ*e*�*θ*1þ*iφ*<sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>2</sup> sin 2*β*2*e*�*i*ð Þ *<sup>φ</sup>*þ*<sup>α</sup> ei* <sup>4</sup>*β*1þ3*β*<sup>2</sup> ð Þ*e*�*θ*2þ*iϕ*<sup>2</sup> <sup>þ</sup>*ρ*<sup>1</sup> sin 2*β*1*e*�*i*ð Þ *<sup>φ</sup>*�*<sup>α</sup> ei* <sup>3</sup>*β*1þ2*β*<sup>2</sup> ð Þ*e*�2*θ*2�*θ*<sup>1</sup> � *ei<sup>ϕ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*<sup>2</sup> sin 2*β*2*ei*ð Þ *<sup>φ</sup>*þ*<sup>α</sup> ei* <sup>2</sup>*β*1þ3*β*<sup>2</sup> ð Þ*e*�2*θ*1�*θ*<sup>2</sup> � *ei<sup>ϕ</sup>*<sup>2</sup> � (81)

where

$$\begin{aligned} \rho &= \arg \left(\lambda\_1^2 - \overline{\lambda\_2^2}\right) = \arctan \left(\rho\_1^2 \sin 2\beta\_1 + \rho\_2^2 \sin 2\beta\_2\right) / \left(\rho\_1^2 \cos 2\beta\_1 - \rho\_2^2 \cos 2\beta\_2\right) \\\ a &= \arg \left(\lambda\_1^2 - \lambda\_2^2\right) = \arctan \left(\rho\_1^2 \sin 2\beta\_1 - \rho\_2^2 \sin 2\beta\_2\right) / \left(\rho\_1^2 \cos 2\beta\_1 - \rho\_2^2 \cos 2\beta\_2\right) \end{aligned} \tag{82}$$

and

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

*<sup>D</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>X<sup>2</sup> *r*¼1 X 1 ≤*n*<sup>1</sup> <*n*<sup>2</sup> ≤2 X 1 ≤ *m*<sup>1</sup> < *m*<sup>2</sup> ≤2 *Q*<sup>1</sup> *n*1*;* ⋯*; nr* ð Þ ; *m*1*;* ⋯*; mr Q*<sup>2</sup> *m*1*;* ⋯*; mr* ð Þ ; *n*1*;* ⋯*; nr* ¼ 1 þ *Q*1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 1 *Q*2ð*m*<sup>1</sup> ¼ 1*; n*<sup>1</sup> ¼ 1Þ þ *Q*1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 2*; n*<sup>1</sup> ¼ 1 þ*Q*1ð Þ *n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 1 *Q*2ð*m*<sup>1</sup> ¼ 1*; n*<sup>1</sup> ¼ 2Þ þ *Q*1ð Þ *n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 2*; n*<sup>1</sup> ¼ 2 þ*Q*1ð Þ *n*<sup>1</sup> ¼ 1*; n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 1*; m*<sup>2</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 1*; m*<sup>2</sup> ¼ 2; *n*<sup>1</sup> ¼ 1*; n*<sup>2</sup> ¼ 2 ¼ 1 � *f* <sup>1</sup> � � � � <sup>4</sup> *λ*<sup>2</sup> 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> � *<sup>f</sup>* <sup>2</sup> � � � � <sup>4</sup> *λ*<sup>2</sup> 2 *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *<sup>f</sup>* 2 1*f* 2 2 *λ*2 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *<sup>f</sup>* 2 1 *f* 2 2 *λ*2 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> þ *f* <sup>1</sup>*f* <sup>2</sup> � � � � 4 � *λ*2 1*λ*2 <sup>2</sup> *λ*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> (83) *D*<sup>2</sup> ¼ 1 þ *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � � � � � � � � � � 2 *e* �*i*2*β*<sup>1</sup> *e* �2*θ*<sup>1</sup> <sup>þ</sup> *<sup>e</sup>* �*i*2*β*<sup>2</sup> *e* �2*θ*<sup>2</sup> � � <sup>þ</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � � � � � � � � � � <sup>2</sup> 0 @ 1 A*e* �ð Þ *<sup>θ</sup>*1þ*θ*<sup>2</sup> *e* �*<sup>i</sup> <sup>β</sup>*1þ*β*<sup>2</sup> ð Þ *<sup>ρ</sup>*<sup>1</sup> *ρ*2 *e <sup>i</sup> <sup>φ</sup>*2�*φ*<sup>1</sup> ð Þ <sup>þ</sup> *<sup>ρ</sup>*<sup>2</sup> *ρ*1 *e <sup>i</sup> <sup>φ</sup>*1�*φ*<sup>2</sup> ð Þ � � þ *e* �*i*<sup>2</sup> *<sup>β</sup>*1þ*β*<sup>2</sup> ð Þ*e* �2ð Þ *θ*1þ*θ*<sup>2</sup> (84)

where

$$\left|\frac{\lambda\_1^2 - \overline{\lambda\_2^2}}{\lambda\_1^2 - \lambda\_2^2}\right|^2 = \frac{\left(\rho\_1/\rho\_2 - \rho\_2/\rho\_1\right)^2 + 4\sin^2(\beta\_1 + \beta\_2)}{\left(\rho\_1/\rho\_2 - \rho\_2/\rho\_1\right)^2 + 4\sin^2(\beta\_1 - \beta\_2)}\tag{85}$$

Substituting (81) and (84) into formula (52), we thus get the two-soliton solution to the DNLS equation with VBC

$$
\overline{\mu}\_2 = -i2A\_2D\_2/\overline{D}\_2^2\tag{86}
$$

Once again we find that, up to a permitted global constant phase factor, the above two-soliton solution is equivalent to that gotten in Ref. [23, 24], verifying the validity of our formula of *N*-soliton solution and the reliability of those linear algebra techniques. As a matter of fact, a general and strict demonstration of our revised IST for DNLS equation with VBC has been given in one paper by use of Liouville theorem [25].

#### **2.5 The asymptotic behaviors of** *N***-soliton solution**

The complex conjugate of expression (52) gives the explicit expression of *N*-soliton solution as

$$
\mu\_N = \text{i}2\overline{A}\_N \overline{D}\_N / D\_N^2 \tag{87}
$$

Without the loss of generality**,** for *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *<sup>μ</sup><sup>n</sup>* <sup>þ</sup> *ivn*,*V*<sup>n</sup> ¼ �<sup>4</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>n</sup>* � *<sup>v</sup>*<sup>2</sup> *n* � �, *n* ¼ 1, 2, ⋯, *N*, we assume *V*<sup>1</sup> <*V*<sup>2</sup> < ⋯ <*Vn* < ⋯*VN* and define the *n*'th vicinity area as Γ*<sup>n</sup>* : *x* � *xno* � *Vnt* � 0,ð Þ *n* ¼ 1*:*2, ⋯, *N* .

As *t* ! �∞, *N* vicinity areas Γ*n*, *n* ¼ 1, 2… , *N*, queue up in a descending series

$$
\Gamma\_N, \Gamma\_{N-1}, \dots, \Gamma\_1 \tag{88}
$$

and in the vicinity of Γ*n*, we have (note that *κ <sup>j</sup>* >0)

$$\theta\_j = 4\kappa\_j(\mathbf{x} - \mathbf{x}\_{j0} - \mathbf{V}\_j \mathbf{t}) \to \begin{cases} ^{+\infty, \quad \text{for } j > n} \\ ^{-\infty, \quad \text{for } j < n} \end{cases} \tag{89}$$

*<sup>θ</sup> <sup>j</sup>* <sup>¼</sup> <sup>4</sup>*<sup>κ</sup> <sup>j</sup> <sup>x</sup>* � *<sup>x</sup> <sup>j</sup>*<sup>0</sup> � *<sup>V</sup> jt* � � ! �∞, for *<sup>j</sup>* <sup>&</sup>gt;*<sup>n</sup>*

4 !

*λ*2 *j* �*λ*<sup>2</sup> *n λ*2 *j* �*λ* 2 *n*

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

So as *t* ! ∞, in the vicinity of Γ*n*,

*<sup>n</sup>* ¼ � <sup>X</sup>

<sup>¼</sup> <sup>2</sup> <sup>X</sup> *N*

**2.6** *N***-soliton solution to MNLS equation**

*N*

*j*¼*n*þ1

*j*¼*n*þ1

Δ*φ*ð Þ <sup>þ</sup>

then as *t* ! ∞,

total backward shift Δ*θ*ð Þ <sup>þ</sup>

Δ*θ*ð Þ �

**37**

� � � �

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup>

*nFn* Q*N j*¼*n*þ1 þ∞, for *j* <*n*

*A* ≃ *AN*�*n*þ<sup>1</sup>ð Þ *n*, *n* þ 1, ⋯, *N*; 0, *n* þ 1, *n* þ 2, ⋯, *N*

*<sup>u</sup>* <sup>¼</sup> *<sup>i</sup>*2*AD=D*<sup>2</sup> <sup>≃</sup>*u*<sup>1</sup> *<sup>θ</sup><sup>n</sup>* <sup>þ</sup> <sup>Δ</sup>*θ*ð Þ <sup>þ</sup>

*<sup>n</sup>* <sup>¼</sup> <sup>2</sup> <sup>X</sup>

*λ*2 *<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � �<sup>2</sup>

*λ*2 *<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � �<sup>2</sup>

*<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � � � arg *<sup>λ</sup>*<sup>2</sup>

*u*<sup>1</sup> *θ<sup>n</sup>* þ Δ*θ*ð Þ <sup>þ</sup>

That is to say, the *N*-soliton solution can be viewed as *N* well-separated exact one- solitons, queuing up in a series with ascending order number *n*: Γ1, Γ2, ⋯, Γ*N:* In the course going from *t* ! �∞ to *t* ! ∞, the *n*'th one-soliton overtakes the solitons from the first to *n* � 1'th and is overtaken by the solitons from *n* þ 1'th to *N*'th. In the meantime, due to collisions, the *n*'th soliton got a total forward shift

*<sup>n</sup> =κ<sup>n</sup>* from exceeding those slower soliton from the first to *n* � 1'th, and got a

*<sup>n</sup>* � Δ*θ*ð Þ � *n*

*<sup>n</sup>* � Δ*φ*ð Þ �

� �

Finally, we indicate that the exact *N*-soliton solution to the DNLS equation can be converted to that of MNLS equation by a gauge-like transformation. A nonlinear

*n* þ 1'th to *N*'th, and just equals to the summation of shifts due to each collision

between two solitons, together with a total phase shift Δ*φn*, that is,

Δ*xn* ¼ Δ*θ*ð Þ <sup>þ</sup>

�

Δ*φ<sup>n</sup>* ¼ Δ*φ*ð Þ <sup>þ</sup>

Schrödinger equation including the nonlinear dispersion term expressed as

*N*

*j*¼*n*þ1

ln *λ*2 *<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*

3 7 <sup>5</sup> <sup>þ</sup> <sup>4</sup>*<sup>β</sup> <sup>j</sup>*

*λ*2 *<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*

� � � � �

Δ*θ*ð Þ <sup>þ</sup>

arg

arg *λ*<sup>2</sup>

8 ><

>:

*uN* <sup>≃</sup> <sup>X</sup> *N*

*n*¼1

2 6 4 �*θ<sup>n</sup> e* �*iφ<sup>n</sup>*

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

*D* ≃ *DN*�*n*ð*n* þ 1, *n* þ 2, ⋯, *N*Þ þ *DN*�*n*þ1ð Þ *n*, *n* þ 1, ⋯, *N*

*λ*2 *j λ* 2 *j*

, (98)

(100)

*DN*�*n*ð Þ *<sup>n</sup>* <sup>þ</sup> 1, *<sup>n</sup>* <sup>þ</sup> 2, <sup>⋯</sup>, *<sup>N</sup>* (97)

*λ*2 *<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � �<sup>2</sup>

*λ*2 *<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � �<sup>2</sup>

*<sup>n</sup>* , *φ<sup>n</sup>* þ Δ*φ*ð Þ <sup>þ</sup>

� � � � �

9 >=

>;

*<sup>j</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � � � <sup>2</sup>*<sup>β</sup> <sup>j</sup>* h i, (101)

*<sup>n</sup>* , *φ<sup>n</sup>* þ Δ*φ*ð Þ <sup>þ</sup>

*<sup>n</sup> =κ<sup>n</sup>* from being exceeded by those faster solitons from

�*=κ<sup>n</sup>* (103)

*<sup>n</sup>* (104)

*n* � � (102)

2

*n* � � (99)

Y *N*

*j*¼*n*þ1

n

¼ *DN*�*<sup>n</sup>*ð Þ *n* þ 1, *n* þ 2, ⋯, *N e*

� � � �

Here the complex constant 2*cn*<sup>0</sup> in expression (65) has been absorbed into *e*�*θ<sup>n</sup> e<sup>i</sup>φ<sup>n</sup>* by redefinition of the soliton center *xn*<sup>0</sup> and the initial phase *αn*0.

Introducing a typical factor *Fn* ¼ �*e*�2*θ<sup>n</sup> <sup>=</sup> <sup>λ</sup>*<sup>2</sup> *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � �<sup>2</sup> >0, *n* ¼ 1, 2, ⋯, *N*; then

$$D\_n(1,2,\cdots,n) = \prod\_{j=1}^n \overline{\lambda\_j^2} F\_j \prod\_{l$$

*where l*, *m* ∈ f g 1, 2, ⋯, *n* . Thus

$$D \simeq D\_{n-1}(\mathbf{1}, \mathbf{2}, \cdots, n-1) + D\_n(\mathbf{1}, \mathbf{2}, \cdots, n) = \left(\mathbf{1} + \overline{\lambda\_n^2} F\_n \prod\_{j=1}^{n-1} \left| \frac{\lambda\_j^2 - \lambda\_n^2}{\lambda\_j^2 - \overline{\lambda}\_n^2} \right|^4 \right) D\_{n-1}(\mathbf{1}, \mathbf{2}, \cdots, n-1) \tag{91}$$

and

$$\overline{A} \simeq \overline{A}\_n(1, 2, \cdots, n; \, 0, 1, 2, \cdots n - 1) = D\_{n-1} e^{-\theta\_n} e^{-i\rho\_n} \prod\_{j=1}^{n-1} \frac{\left(\overline{\lambda\_j^2} - \overline{\lambda\_n^2}\right)^2}{\left(\lambda\_j^2 - \overline{\lambda\_n^2}\right)^2} e^{i4\theta\_j} \tag{92}$$

In the vicinity of Γ*n*,

$$u(\mathbf{x},t) = i2\overline{\mathbf{A}}\overline{\mathbf{D}}/\mathbf{D}^2 \simeq u\_1\left(\theta\_n + \Delta\theta\_n^{(-)}, \rho\_n + \Delta\rho\_n^{(-)}\right) \tag{93}$$

Here

$$\Delta\theta\_n^{(-)} = 2\sum\_{j=1}^{n-1} \ln\left|\frac{\lambda\_j^2 - \overline{\lambda\_n^2}}{\lambda\_j^2 - \overline{\lambda\_n^2}}\right|^2 \tag{94}$$

$$\begin{split} \Delta\rho\_n^{(-)} &= -\sum\_{j=1}^{n-1} \left\{ \arg\left[\frac{\left(\overline{\lambda\_j^2} - \overline{\lambda\_n^2}\right)^2}{\left(\lambda\_j^2 - \overline{\lambda\_n^2}\right)^2}\right] + 4\beta\_j \right\} \\ &= 2\sum\_{j=1}^{n-1} \left[ \arg\left(\lambda\_j^2 - \overline{\lambda\_n^2}\right) - \arg\left(\overline{\lambda\_j^2} - \overline{\lambda\_n^2}\right) - 2\beta\_j \right] \end{split} \tag{95}$$

then

$$u\_N \simeq \sum\_{n=1}^N u\_1 \left(\theta\_n + \Delta \theta\_n^{(-)}, \rho\_n + \Delta \rho\_n^{(-)}\right) \tag{96}$$

Each *u*<sup>1</sup> *θn*, *φ<sup>n</sup>* ð Þ, (1, 2, ⋯, *n*) is a one-soliton solution characterized by one parameter *λn*, moving along the positive direction of the x-axis, queuing up in a series with descending order number *n* as in series (88). As *t* ! ∞, in the vicinity of Γ*n*, we have (note that *κ <sup>j</sup>* >0)

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

$$\begin{aligned} \boldsymbol{\theta}\_{j} &= 4\kappa\_{j} \{ \mathbf{x} - \mathbf{x}\_{\,j0} - \mathbf{V}\_{\,j} \mathbf{t} \} \ & \quad \underset{\mathbf{x} \sim \mathbf{x}\_{\,j}}{\operatorname{ for } j < n} \quad D \simeq D\_{N-n} (\mathfrak{n} + \mathbf{1}, \mathfrak{n} + \mathbf{2}, \dots, \mathbf{N}) + D\_{N-n+1} (\mathfrak{n}, \mathfrak{n} + \mathbf{1}, \dots, \mathfrak{N}) \\\\ & \quad \left\langle \begin{array}{cccc} & & \mathbf{1} & & \mathbf{1} & & \mathbf{1} & & \mathbf{1} \end{array} \right\rangle \end{aligned}$$

$$\mathbf{x} = \left(\mathbf{1} + \overline{\lambda\_n^2} F\_n \prod\_{j=n+1}^{N} \left| \frac{\mathbf{i}\_j^2 - \overline{\mathbf{i}\_n^2}}{\mathbf{i}\_j^2 - \overline{\mathbf{i}\_n}} \right|^4 \right) D\_{N-n}(n+1, n+2, \cdots, N) \tag{97}$$

$$\begin{split} \mathcal{A} & \simeq A\_{N-n+1}(n, n+1, \cdots, N; \mathbf{0}, n+1, n+2, \cdots, N) \\ &= D\_{N-n}(n+1, n+2, \cdots, N) e^{-\theta\_n} e^{-i\rho\_n} \prod\_{j=n+1}^{N} \frac{\left(\overline{\lambda\_j^2} - \overline{\lambda\_n^2}\right)^2 \lambda\_j^2}{\left(\lambda\_j^2 - \overline{\lambda\_n^2}\right)^2 \overline{\lambda\_j^2}}, \end{split} \tag{98}$$

So as *t* ! ∞, in the vicinity of Γ*n*,

$$\mu = i2\overline{AD}/D^2 \simeq u\_1 \left(\theta\_n + \Delta \theta\_n^{(+)}, \rho\_n + \Delta \rho\_n^{(+)}\right) \tag{99}$$

$$\Delta\theta\_n^{(+)} = 2\sum\_{j=n+1}^{N} \ln \left| \frac{\lambda\_j^2 - \overline{\lambda\_n^2}}{\lambda\_j^2 - \lambda\_n^2} \right|^2 \tag{100}$$

$$\begin{split} \Delta \rho\_{n}^{(+)} &= -\sum\_{j=n+1}^{N} \left\{ \arg \left[ \frac{\left(\overline{\lambda\_{j}^{2}} - \overline{\lambda\_{n}^{2}}\right)^{2}}{\left(\lambda\_{j}^{2} - \overline{\lambda\_{n}^{2}}\right)^{2}} \right] + 4\beta\_{j} \right\} \\ &= 2 \sum\_{j=n+1}^{N} \left[ \arg \left(\lambda\_{j}^{2} - \overline{\lambda\_{n}^{2}}\right) - \arg \left(\overline{\lambda\_{j}^{2}} - \overline{\lambda\_{n}^{2}}\right) - 2\beta\_{j} \right], \end{split} \tag{101}$$

then as *t* ! ∞,

$$u\_N \simeq \sum\_{n=1}^N u\_1 \left(\theta\_n + \Delta \theta\_n^{(+)}, \rho\_n + \Delta \rho\_n^{(+)}\right) \tag{102}$$

That is to say, the *N*-soliton solution can be viewed as *N* well-separated exact one- solitons, queuing up in a series with ascending order number *n*: Γ1, Γ2, ⋯, Γ*N:* In the course going from *t* ! �∞ to *t* ! ∞, the *n*'th one-soliton overtakes the solitons from the first to *n* � 1'th and is overtaken by the solitons from *n* þ 1'th to *N*'th. In the meantime, due to collisions, the *n*'th soliton got a total forward shift Δ*θ*ð Þ � *<sup>n</sup> =κ<sup>n</sup>* from exceeding those slower soliton from the first to *n* � 1'th, and got a total backward shift Δ*θ*ð Þ <sup>þ</sup> *<sup>n</sup> =κ<sup>n</sup>* from being exceeded by those faster solitons from *n* þ 1'th to *N*'th, and just equals to the summation of shifts due to each collision between two solitons, together with a total phase shift Δ*φn*, that is,

$$
\Delta \mathfrak{x}\_n = \left| \Delta \theta\_n^{(+)} - \Delta \theta\_n^{(-)} \right| / \kappa\_n \tag{103}
$$

$$
\Delta \rho\_n = \Delta \rho\_n^{(+)} - \Delta \rho\_n^{(-)} \tag{104}
$$

#### **2.6** *N***-soliton solution to MNLS equation**

Finally, we indicate that the exact *N*-soliton solution to the DNLS equation can be converted to that of MNLS equation by a gauge-like transformation. A nonlinear Schrödinger equation including the nonlinear dispersion term expressed as

$$i\partial\_t \nu + \partial\_{\text{xx}} \nu + ia \partial\_{\text{x}} \left( \left| \nu \right|^2 \nu \right) + 2\beta \left| \nu \right|^2 \nu = \mathbf{0} \tag{105}$$

**3.1 The lax pair and its Jost functions of DNLS equation**

*<sup>L</sup>* ¼ �*iλ*<sup>2</sup>

*<sup>M</sup>* ¼ �*i*2*λ*<sup>4</sup>*σ*<sup>3</sup> <sup>þ</sup> <sup>2</sup>*λ*<sup>3</sup>

In the case of <sup>∣</sup>*x*<sup>∣</sup> ! <sup>∞</sup>, *<sup>u</sup>* ! 0, *<sup>L</sup>* ! *<sup>L</sup>*<sup>0</sup> ¼ �*iλ*<sup>2</sup>

�*iλ*2*xσ*<sup>3</sup> , *<sup>E</sup>*•1ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>*

**3.2 The integral representation of Jost function**

*N x*ð Þ , *y* <sup>11</sup> 0

0 *N x*ð Þ , *y* <sup>22</sup> � �, *<sup>N</sup><sup>o</sup>*

Due to the symmetry of the first Lax operator *λ*<sup>2</sup>

function should have the same symmetry as follows:

*<sup>N</sup><sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>11</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup>

Ψð Þ¼ *x*, *λ E x*ð Þþ , *λ*

suppressed temporarily. Here

*λ*2

*<sup>N</sup><sup>d</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*

**39**

As usual, we introduce the integral representation,

ð<sup>∞</sup> *x* d*y λ*<sup>2</sup>

*iut* <sup>þ</sup> *uxx* <sup>þ</sup> *i u*j j<sup>2</sup>

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

with vanishing boundary, ∣*x*∣ ! ∞, *u* ! 0. Here the subscript denotes partial

*<sup>U</sup>* � *<sup>i</sup>λ*<sup>2</sup>

1 0 � �*<sup>e</sup>*

where *<sup>λ</sup>*<sup>2</sup> is a real squared parameter, *E x*ð Þ , *<sup>λ</sup>* expresses two independent solutions with two components. The Jost solutions of (4) are defined by their

*u* � �

*<sup>σ</sup>*<sup>3</sup> <sup>þ</sup> *<sup>λ</sup>U*, *<sup>U</sup>* <sup>¼</sup> <sup>0</sup> *<sup>u</sup>*

*U*2

�*u* 0

*<sup>σ</sup>*<sup>3</sup> � *<sup>λ</sup>* �*U*<sup>3</sup> <sup>þ</sup> *iUxσ*<sup>3</sup>

*<sup>∂</sup>xf x*ð Þ¼ , *<sup>λ</sup> L x*ð Þ , *<sup>λ</sup> f x*ð Þ , *<sup>λ</sup>* (111)

�*iλ*2*<sup>x</sup>*; *<sup>E</sup>*•2ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>*

Ψð Þ¼ *x*, *λ* ð*ψ*~ð Þ *x*, *λ* , *ψ*ð Þ *x*, *λ* Þ ! *E x*ð Þ , *λ* , *as x* ! ∞ (113) Φð Þ¼ *x*, *λ* ð*φ*ð Þ *x*, *λ* , *φ*~ð Þ *x*, *λ* Þ ! *E x*ð Þ , *λ* , *as x* ! �∞ (114)

*<sup>N</sup><sup>d</sup>*ð Þþ *<sup>x</sup>*, *<sup>y</sup> <sup>λ</sup>N<sup>o</sup>*

ð Þ¼ *x*, *y*

ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>21</sup> ¼ �*λN<sup>o</sup>*

where the superscripts d and o mean the diagonal and off-diagonal elements, respectively. According to the conventional operation in IST, the time variable is

*λU*<sup>21</sup> ¼ �*λU*12, the kernel matrix *N x*ð Þ , *y* of the integral representation of Jost

*<sup>N</sup><sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>22</sup>; *<sup>λ</sup>N<sup>o</sup>*

*σ*3, the free Jost solution is

0 1 � �*<sup>e</sup>*

<sup>ð</sup>*x*, *<sup>y</sup>*Þg*E y*ð Þ , *<sup>λ</sup>* � (115)

0 *N x*ð Þ , *y* <sup>12</sup>

ð Þ �*iσ*<sup>3</sup> <sup>22</sup> and

ð Þ *x*, *y* <sup>12</sup> (116)

*N x*ð Þ , *y* <sup>21</sup> 0 � �

ð Þ �*iσ*<sup>3</sup> <sup>11</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup>

!

*<sup>x</sup>* <sup>¼</sup> <sup>0</sup> (108)

� � (110)

(109)

*<sup>i</sup>λ*2*<sup>x</sup>* (112)

DNLS equation is usual expressed as

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

derivative. Its Lax pair is given by

The first Lax equation is

*E x*ð Þ¼ , *λ e*

asymptotic properties at *x* ! �∞,

is also integrable [23] and called modified nonlinear Schrödinger (MNLS for brevity) equation. It is well known that MNLS equation well describes transmission of femtosecond pulses in optical fibers [4–6] and is related to DNLS equation by a gauge-like transformation [23] formulated as

$$\nu(\mathbf{x},t) = \mathfrak{u}(X,T)e^{i2\rho X + i4\rho^2 T} \tag{106}$$

with *<sup>x</sup>* <sup>¼</sup> *<sup>α</sup>*�<sup>1</sup>ð Þ *<sup>X</sup>* <sup>þ</sup> <sup>4</sup>*ρ<sup>T</sup>* , *<sup>t</sup>* <sup>¼</sup> *<sup>α</sup>*�<sup>2</sup>*T*; *<sup>X</sup>* <sup>¼</sup> *<sup>α</sup><sup>x</sup>* � <sup>4</sup>*βt*, *<sup>T</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>2</sup>*t*; *<sup>ρ</sup>* <sup>¼</sup> *βα*�<sup>2</sup> . Using a method that is analogous to reference [16], and applying above gauge-like transformation to Eq. (105), the MNLS equation with VBC can be transformed into DNLS equation with VBC.

$$i\partial\_T u + \partial\_{\text{XX}} u + i\partial\_{\text{X}} \left( |u|^2 u \right) = \mathbf{0} \tag{107}$$

with *u* ¼ *u X*ð Þ , *T* . So according to (106), the *N*-soliton solution to MNLS equation can also be attained by a gauge-like transformation from that of DNLS equation.

The *N*-soliton solution to the DNLS equation with VBC has been derived by means of a IST considered anew and some special linear algebra techniques. The one- and two-soliton solutions have been given as two typical examples in illustration of the general formula of the *N*-soliton solution. It is found to be perfectly in agreement with that gotten in the following section based on a pure Marchenko formalism or Hirota's Bilinear derivative transformation [24, 26, 27]. The demonstration of the revised IST considered anew for DNLS equation with VBC has also been given by use of Liouville theorem [25].

The newly revised IST technique for DNLS equation with VBC supplies substantial foundation for its direct perturbation theory.

## **3. A simple method to derive and solve Marchenko equation for DNLS equation**

Gel'fand-Levitan-Marchenko (GLM for brevity) equations can be viewed as an integral-transformed version of IST for those integrable nonlinear equations [21, 24, 28].

In this section, a simple method is used to derive and solve Marchenko equation (or GLM equation) for DNLS E with VBC [28]. Firstly, starting from the first Lax equation, we derive two conditions to be satisfied by the kernel matrix *N x*ð Þ , *y* of GLM by applying the Lax operator *<sup>∂</sup><sup>x</sup>* � *<sup>L</sup>* upon the integral representation of Jost function for DNLSE. Secondly, based on Lax equation, a strict demonstration has been given for the validness of Marchenko formalism. At last, the Marchenko formalism is determined by choosing a suitable *F x*ð Þ þ *y* and *G x*ð Þ þ *y* , and their relation (135) has been constructed. The one and multi-soliton solution in the reflectionless case is attained based upon a pure Marchenko formalism by avoiding direct use of inverse scattering data and verified by using direct substitution method with Mathematica.

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

## **3.1 The lax pair and its Jost functions of DNLS equation**

DNLS equation is usual expressed as

$$\left(\dot{u}\_t + u\_{\text{xx}} + i\left(|u|^2 u\right)\_{\text{x}} = \mathbf{0}\right) \tag{108}$$

with vanishing boundary, ∣*x*∣ ! ∞, *u* ! 0. Here the subscript denotes partial derivative. Its Lax pair is given by

$$L = -i\lambda^2 \sigma\_3 + \lambda U,\\ U = \begin{pmatrix} 0 & u \\ -\overline{u} & 0 \end{pmatrix} \tag{109}$$

$$M = -i2\lambda^4 \sigma\_3 + 2\lambda^3 U - i\lambda^2 U^2 \sigma\_3 - \lambda \left(-U^3 + iU\_x \sigma\_3\right) \tag{110}$$

The first Lax equation is

$$
\partial\_{\mathbf{x}} f(\mathbf{x}, \boldsymbol{\lambda}) = L(\mathbf{x}, \boldsymbol{\lambda}) f(\mathbf{x}, \boldsymbol{\lambda}) \tag{111}
$$

In the case of <sup>∣</sup>*x*<sup>∣</sup> ! <sup>∞</sup>, *<sup>u</sup>* ! 0, *<sup>L</sup>* ! *<sup>L</sup>*<sup>0</sup> ¼ �*iλ*<sup>2</sup> *σ*3, the free Jost solution is

$$E(\mathbf{x}, \boldsymbol{\lambda}) = e^{-i\boldsymbol{\lambda}^2 \mathbf{x} \boldsymbol{\sigma} \boldsymbol{\lambda}}, E\_1(\mathbf{x}, \boldsymbol{\lambda}) = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix} e^{-i\boldsymbol{\lambda}^2 \mathbf{x}}; E\_2(\mathbf{x}, \boldsymbol{\lambda}) = \begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix} e^{i\boldsymbol{\lambda}^2 \mathbf{x}} \tag{112}$$

where *<sup>λ</sup>*<sup>2</sup> is a real squared parameter, *E x*ð Þ , *<sup>λ</sup>* expresses two independent solutions with two components. The Jost solutions of (4) are defined by their asymptotic properties at *x* ! �∞,

$$\Psi(\mathbf{x},\lambda) = (\tilde{\boldsymbol{\mu}}(\mathbf{x},\lambda), \boldsymbol{\mu}(\mathbf{x},\lambda)) \to E(\mathbf{x},\lambda), \quad \text{as} \ \boldsymbol{\pi} \to \infty \tag{113}$$

$$\Phi(\mathbf{x},\lambda) = (\varphi(\mathbf{x},\lambda), \bar{\varphi}(\mathbf{x},\lambda)) \to E(\mathbf{x},\lambda), \text{ as } \mathbf{x} \to -\infty \tag{114}$$

#### **3.2 The integral representation of Jost function**

As usual, we introduce the integral representation,

$$\Psi(\mathbf{x},\lambda) = E(\mathbf{x},\lambda) + \int\_{\mathbf{x}}^{\mathbf{os}} \mathrm{d}\mathfrak{y} \{\lambda^2 \mathcal{N}^d(\mathbf{x},\mathfrak{y}) + \lambda \mathcal{N}^o(\mathbf{x},\mathfrak{y})\} E(\mathfrak{y},\lambda) \tag{115}$$

where the superscripts d and o mean the diagonal and off-diagonal elements, respectively. According to the conventional operation in IST, the time variable is suppressed temporarily. Here

$$N^d(\mathbf{x}, \boldsymbol{\mathcal{y}}) = \begin{pmatrix} N(\mathbf{x}, \boldsymbol{\mathcal{y}})\_{11} & \mathbf{0} \\ \mathbf{0} & N(\mathbf{x}, \boldsymbol{\mathcal{y}})\_{22} \end{pmatrix}, \\ N^\boldsymbol{\sigma}(\mathbf{x}, \boldsymbol{\mathcal{y}}) = \begin{pmatrix} \mathbf{0} & N(\mathbf{x}, \boldsymbol{\mathcal{y}})\_{12} \\ N(\mathbf{x}, \boldsymbol{\mathcal{y}})\_{21} & \mathbf{0} \end{pmatrix}$$

Due to the symmetry of the first Lax operator *λ*<sup>2</sup> ð Þ �*iσ*<sup>3</sup> <sup>11</sup> <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup> ð Þ �*iσ*<sup>3</sup> <sup>22</sup> and *λU*<sup>21</sup> ¼ �*λU*12, the kernel matrix *N x*ð Þ , *y* of the integral representation of Jost function should have the same symmetry as follows:

$$
\lambda^2 N^d(\mathbf{x}, \mathbf{y})\_{11} = \lambda^2 \overline{N^d(\mathbf{x}, \mathbf{y})\_{22}}; \\
\lambda N^o(\mathbf{x}, \mathbf{y})\_{21} = -\lambda \overline{N^o(\mathbf{x}, \mathbf{y})\_{12}} \tag{116}
$$

Substitute Eq. (115) into the first Lax Eq. (111). By simply partial integration, we have the following terms:

$$\{\partial\_{\mathbf{x}} - L\} E(\mathbf{x}, \lambda) = -(L - L\_0) E(\mathbf{x}, \lambda) = -\lambda U(\mathbf{x}) E(\mathbf{x}, \lambda) \tag{117}$$

*∂x* Ð <sup>∞</sup> *<sup>x</sup>* <sup>d</sup>*<sup>y</sup> <sup>λ</sup>*<sup>2</sup> *<sup>N</sup><sup>d</sup>*ð Þþ *<sup>x</sup>; <sup>y</sup> <sup>λ</sup>N<sup>o</sup>* <sup>ð</sup>*x; <sup>y</sup>*<sup>Þ</sup> � �*E y*ð Þ *; <sup>λ</sup>* ¼ � *<sup>λ</sup>*<sup>2</sup> *<sup>N</sup><sup>d</sup>*ð Þþ *<sup>x</sup>; <sup>x</sup> <sup>λ</sup>N<sup>o</sup>* <sup>ð</sup>*x; <sup>x</sup>*<sup>Þ</sup> � �*E x*ð Þþ *; <sup>λ</sup>* <sup>Ð</sup> <sup>∞</sup> *<sup>x</sup>* <sup>d</sup>*<sup>y</sup> <sup>λ</sup>*<sup>2</sup> *N<sup>d</sup> <sup>x</sup>*ð Þþ *<sup>x</sup>; <sup>y</sup> <sup>λ</sup>N<sup>o</sup> <sup>x</sup>*ð*x; <sup>y</sup>*<sup>Þ</sup> � �*E y*ð Þ *; <sup>λ</sup>* (118) � �*iλ*<sup>2</sup> *σ*3 � �Ð *<sup>x</sup>*d*<sup>y</sup> <sup>λ</sup>*<sup>2</sup> *<sup>N</sup>d*ð Þþ *<sup>x</sup>; <sup>y</sup> <sup>λ</sup>N*<sup>∘</sup> <sup>ð</sup>*x; <sup>y</sup>*<sup>Þ</sup> � �*E y*ð Þ¼� *; <sup>λ</sup>* <sup>Ð</sup> *<sup>x</sup>*d*yσ*<sup>3</sup> *<sup>λ</sup>*<sup>2</sup> *<sup>N</sup>d*ð Þþ *<sup>x</sup>; <sup>y</sup> <sup>λ</sup>N<sup>o</sup>* <sup>ð</sup>*x; <sup>y</sup>*<sup>Þ</sup> � �*σ*3*E*<sup>0</sup> ð Þ *y; λ* <sup>¼</sup> *<sup>σ</sup>*<sup>3</sup> *<sup>λ</sup>*<sup>2</sup> *<sup>N</sup>d*ð Þþ *<sup>x</sup>; <sup>x</sup> <sup>λ</sup>N<sup>o</sup>* <sup>ð</sup>*x; <sup>x</sup>*<sup>Þ</sup> � �*σ*3*E x*ð Þþ *; <sup>λ</sup>* <sup>Ð</sup> <sup>∞</sup> *<sup>x</sup>* <sup>d</sup>*yσ*<sup>3</sup> *<sup>λ</sup>*<sup>2</sup> *N<sup>d</sup> <sup>y</sup>* ð Þþ *<sup>x</sup>; <sup>y</sup> <sup>λ</sup>N<sup>o</sup> <sup>y</sup>*ð*x; y*Þ h i*σ*3*E y*ð Þ *; <sup>λ</sup>*

$$(119)$$

**3.3 Marchenko equation for DNLSE and its demonstration**

form of Marchenko equation for DNLSE with VBC is

*<sup>N</sup><sup>d</sup>*ð Þþ *<sup>x</sup>*, *<sup>y</sup>*

ð Þþ *x*, *y F x*ð Þþ þ *y*

needn't involve obviously the function of spectral parameter *λ*.

ð Þ *x*, *x F x*ð Þþ þ *y*

ð Þ *x*, *x F x*ð Þ� þ *y*

ð Þ� *<sup>x</sup>; <sup>y</sup> <sup>N</sup><sup>o</sup>*

ð<sup>∞</sup> *x* d*zN<sup>o</sup>*

*<sup>y</sup>* ð Þþ *x*, *y*

*Nd*

By partial integrating, Eq. (131) becomes

ð<sup>∞</sup> *x* d*zN<sup>o</sup>*

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

ð<sup>∞</sup> *x*

where *F x*ð Þ þ *y* is only with off-diagonal terms. *G x*ð Þ þ *y* is considered as another function with only off-diagonal terms. We notice that the Marchenko equation

We now show the kernel *N x*ð Þ , *y* determined by (128) and (129) indeed satisfy the conditions (126) and (127) as long as we choose a suitable form of expression for

Making partial derivation in (128) with respect to *x* and *y*, respectively,

ð<sup>∞</sup> *x* d*zN<sup>o</sup>*

ð<sup>∞</sup> *x* d*zN<sup>o</sup>*

f g *l:h:s: of* ð Þ 20 þ *σ*3f g *l:h:s: of* ð Þ 22 *σ*<sup>3</sup> � *U x*ð Þ� f g *l:h:s: of* ð Þ 19 ¼ 0

ð Þ *<sup>x</sup>; <sup>x</sup> F x*ð Þ� <sup>þ</sup> *<sup>y</sup> <sup>σ</sup>*3*N<sup>o</sup>*

*<sup>z</sup>*ð*x; <sup>z</sup>*Þ*F z*ð Þ <sup>þ</sup> *<sup>y</sup> <sup>σ</sup>*<sup>3</sup> � *U x*ð Þ*Nd*ð*x; <sup>z</sup>*Þ*G z*ð Þ <sup>þ</sup> *<sup>y</sup>* � � <sup>¼</sup> <sup>0</sup>

*<sup>z</sup>* ð Þ *<sup>x</sup>; <sup>z</sup> <sup>σ</sup>*3*F z*ð Þ� <sup>þ</sup> *<sup>y</sup> U x*ð Þ*N<sup>d</sup>*ð Þ *<sup>x</sup>; <sup>z</sup> G z*ð Þ <sup>þ</sup> *<sup>y</sup>* � � <sup>¼</sup> <sup>0</sup>

Since *F x*ð Þ is off-diagonal, *F x*ð Þ*σ*<sup>3</sup> ¼ �*σ*3*F x*ð Þ. Thus the terms involving with

*F x*ð Þ <sup>þ</sup> *<sup>y</sup>* outside of integral are equal to �*iU x*ð Þ*N<sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>x</sup> <sup>σ</sup>*3*F x*ð Þ <sup>þ</sup> *<sup>y</sup>* by use of

*G z*ð Þ¼ þ *y iσ*3*F*<sup>0</sup>

ð Þ *x*, *z F*<sup>0</sup>

ð Þ *x*, *y* appear in different manner, we assume the

ð Þ *x*, *z F z*ð Þ¼ þ *y* 0 (128)

<sup>d</sup>*zN<sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>z</sup> G z*ð Þ¼ <sup>þ</sup> *<sup>y</sup>* 0 (129)

*<sup>x</sup>*ð Þ *x*, *z F z*ð Þ¼ þ *y* 0 (130)

ð Þ¼ *z* þ *y* 0 (131)

*<sup>z</sup>*ð Þ *x*, *z F z*ð Þ¼ þ *y* 0 (132)

ð Þ *z* þ *y* , (135)

ð Þ *x; x F x*ð Þ þ *y σ*<sup>3</sup> � *U x*ð Þ*F x*ð Þ þ *y*

(133)

(134)

ð Þ *z* þ *y* = *Fy*ð Þ¼ *z* þ *y Fz*ð Þ *z* þ *y* in (131). Making a

In Eq. (115), the *<sup>N</sup><sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* and *<sup>N</sup><sup>o</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

*No*

*G x*ð Þ þ *y* .

we obtain

*Nd*

*Nd*

We find

*<sup>x</sup>*ð Þþ *<sup>x</sup>; <sup>y</sup> <sup>σ</sup>*3*N<sup>d</sup>*

<sup>þ</sup> <sup>Ð</sup> <sup>∞</sup>

**41**

If we choose

<sup>þ</sup> <sup>Ð</sup> <sup>∞</sup> *<sup>x</sup>* <sup>d</sup>*z N<sup>o</sup>*

*N<sup>d</sup>*

*<sup>x</sup>*ð Þ� *<sup>x</sup>*, *<sup>y</sup> <sup>N</sup><sup>o</sup>*

*<sup>y</sup>* ð Þ� *<sup>x</sup>*, *<sup>y</sup> <sup>N</sup><sup>o</sup>*

Use is made of the fact that *F*<sup>0</sup>

*<sup>y</sup>* ð Þ *<sup>x</sup>; <sup>y</sup> <sup>σ</sup>*<sup>3</sup> � *U x*ð Þ*N<sup>o</sup>*

Eq. (123). Then (133) can be rewritten as

*<sup>x</sup>* <sup>d</sup>*zBx*ð Þ *; <sup>z</sup> F z*ð Þ� <sup>þ</sup> *<sup>y</sup> iU x*ð Þ*N<sup>d</sup>*

*A x*ð Þ� *; <sup>y</sup> iU x*ð Þ*N<sup>d</sup>*ð Þ *<sup>x</sup>; <sup>x</sup> <sup>σ</sup>*3*F x*ð Þ <sup>þ</sup> *<sup>y</sup>*

*<sup>x</sup>*ð Þ *<sup>x</sup>; <sup>z</sup> F z*ð Þ� <sup>þ</sup> *<sup>y</sup> <sup>σ</sup>*3*N<sup>o</sup>*

weighing summation as follows:

and

$$\begin{split} & -\lambda U(\boldsymbol{x}) \int\_{\boldsymbol{x}}^{\infty} \mathsf{d}y \left\{ \lambda^{2} N^{d}(\boldsymbol{x}, \boldsymbol{y}) \right\} E(\boldsymbol{y}, \lambda) = -\int\_{\boldsymbol{x}}^{\infty} \mathsf{d}y \lambda U(\boldsymbol{x}) N^{d}(\boldsymbol{x}, \boldsymbol{y}) i\sigma\_{3} E'(\boldsymbol{y}, \lambda) \\ & = i\lambda U(\boldsymbol{x}) N^{d}(\boldsymbol{x}, \boldsymbol{x}) \sigma\_{3} E(\boldsymbol{x}, \lambda) + i \int\_{\boldsymbol{x}}^{\infty} \mathsf{d}y \lambda U(\boldsymbol{x}) N^{d}\_{\boldsymbol{y}}(\boldsymbol{x}, \boldsymbol{y}) \sigma\_{3} E(\boldsymbol{y}, \lambda) \end{split} \tag{120}$$

Use is made of that �*iλ*<sup>2</sup> *σ*3*E y*ð Þ¼ , *λ E*<sup>0</sup> ð Þ *y*, *λ* , then

$$-\lambda U(\mathbf{x}) \int\_{\mathbf{x}}^{\infty} \mathbf{d}y N^{\sigma}(\mathbf{x}, y) \equiv -\int\_{\mathbf{x}}^{\infty} \mathbf{d}y \lambda U(\mathbf{x}) N^{\sigma}(\mathbf{x}, y) \tag{121}$$

According to equation ð Þ *<sup>∂</sup><sup>x</sup>* � *<sup>L</sup>* <sup>Ψ</sup>ð Þ¼ *<sup>x</sup>*, *<sup>λ</sup>* 0, adding up the l.h.s. and r.h.s., respectively, of Eq. (117)–(120), (121). We obtain two equations involving with terms *λ*<sup>2</sup> and *λ* outside of the integral Ð d*y*⋯ as follows:

$$\lambda^2: -N^d(\mathfrak{x}, \mathfrak{x}) + \sigma\_3 N^d(\mathfrak{x}, \mathfrak{x}) \sigma\_3 = \mathbf{0} \tag{122}$$

$$\lambda^1: -U(\mathbf{x}) - \mathbf{N}^o(\mathbf{x}, \mathbf{x}) + \sigma\_3 \mathbf{N}^o(\mathbf{x}, \mathbf{x})\sigma\_3 + iU(\mathbf{x})\mathbf{N}^d(\mathbf{x}, \mathbf{x})\sigma\_3 = \mathbf{0} \tag{123}$$

Or

$$U\_{12} = \mathfrak{u}(\mathfrak{x}) = -2 \frac{N\_{12}(\mathfrak{x}, \mathfrak{x})}{\mathbf{1} + i \overline{N}\_{11}(\mathfrak{x}, \mathfrak{x})},\tag{124}$$

and the equations in the integral Ð d*y S*f g ¼ 0, where f g*S* is equal to

$$\begin{bmatrix} \lambda^2 N\_\mathbf{x}^d(\mathbf{x}, \mathbf{y}) + \lambda N\_\mathbf{x}^p(\mathbf{x}, \mathbf{y}) \end{bmatrix} + \sigma\_3 \begin{bmatrix} \lambda^2 N\_\mathbf{y}^d(\mathbf{x}, \mathbf{y}) + \lambda N\_\mathbf{y}^p(\mathbf{x}, \mathbf{y}) \end{bmatrix} \sigma\_3 - \lambda U(\mathbf{x}) \begin{bmatrix} -i N\_\mathbf{y}^d(\mathbf{x}, \mathbf{y}) \sigma\_3 + \lambda N^p(\mathbf{x}, \mathbf{y}) \end{bmatrix} = \mathbf{0} \tag{125}$$

Therefore, Eq. (125) gives two conditions to be satisfied by the kernel matrix *N x*ð Þ , *y* in the integral representation of Jost solution

$$\lambda^2 \text{ terms}: A(\mathbf{x}, \mathbf{y}) \equiv \mathbf{N}\_x^d(\mathbf{x}, \mathbf{y}) + \sigma\_3 \mathbf{N}\_y^d(\mathbf{x}, \mathbf{y}) \sigma\_3 - U(\mathbf{x}) \mathbf{N}^p(\mathbf{x}, \mathbf{y}) = \mathbf{0} \tag{126}$$

$$\lambda^1 \text{ terms}: B(\mathbf{x}, \mathbf{y}) \equiv N\_\mathbf{x}^\rho(\mathbf{x}, \mathbf{y}) + \sigma\_3 N\_\mathbf{y}^\rho(\mathbf{x}, \mathbf{y}) \sigma\_3 + iU(\mathbf{x}) N\_\mathbf{y}^d(\mathbf{x}, \mathbf{y}) \sigma\_3 = \mathbf{0} \tag{127}$$

Since (122) is an identity, Eq. (123) or (124) gives the solution *U x*ð Þ or *u x*ð Þ in terms of *N*(*x*,*x*), thus the first Lax equation gives two conditions (126) and (127) which should be satisfied by the integral kernel *N*(*x*,*y*). Note that the time variable of *u x*ð Þ in (124) is suppressed temporarily.

## **3.3 Marchenko equation for DNLSE and its demonstration**

In Eq. (115), the *<sup>N</sup><sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* and *<sup>N</sup><sup>o</sup>* ð Þ *x*, *y* appear in different manner, we assume the form of Marchenko equation for DNLSE with VBC is

$$N^d(\mathbf{x}, \boldsymbol{\mathcal{y}}) + \int\_{\boldsymbol{\mathcal{x}}}^{\boldsymbol{\infty}} \mathrm{d}z N^{\boldsymbol{\nu}}(\mathbf{x}, \boldsymbol{z}) F(\boldsymbol{z} + \boldsymbol{\mathcal{y}}) = \mathbf{0} \tag{128}$$

$$N^{\theta}(\mathbf{x}, \boldsymbol{\uprho}) + F(\mathbf{x} + \boldsymbol{\uprho}) + \int\_{\boldsymbol{\uprho}}^{\infty} \mathbf{d} \mathbf{z} \mathbf{N}^{d}(\mathbf{x}, \boldsymbol{z}) \mathbf{G}(\mathbf{z} + \boldsymbol{\uprho}) = \mathbf{0} \tag{129}$$

where *F x*ð Þ þ *y* is only with off-diagonal terms. *G x*ð Þ þ *y* is considered as another function with only off-diagonal terms. We notice that the Marchenko equation needn't involve obviously the function of spectral parameter *λ*.

We now show the kernel *N x*ð Þ , *y* determined by (128) and (129) indeed satisfy the conditions (126) and (127) as long as we choose a suitable form of expression for *G x*ð Þ þ *y* .

Making partial derivation in (128) with respect to *x* and *y*, respectively, we obtain

$$\mathrm{N}\_{\mathrm{x}}^{\mathrm{ul}}(\boldsymbol{\mathfrak{x}},\boldsymbol{\mathfrak{y}}) - \mathrm{N}^{\boldsymbol{\sigma}}(\boldsymbol{\mathfrak{x}},\boldsymbol{\mathfrak{x}})F(\boldsymbol{\mathfrak{x}}+\boldsymbol{\mathfrak{y}}) + \int\_{\mathrm{x}}^{\mathrm{os}} \mathrm{d} \mathrm{z} \mathrm{N}\_{\mathrm{x}}^{\boldsymbol{\sigma}}(\boldsymbol{\mathfrak{x}},\boldsymbol{\mathfrak{z}})F(\boldsymbol{\mathfrak{z}}+\boldsymbol{\mathfrak{y}}) = \mathbf{0} \tag{130}$$

$$N\_{\mathcal{I}}^d(\boldsymbol{\omega}, \boldsymbol{\jmath}) + \int\_{\boldsymbol{\omega}}^{\infty} \mathrm{d}z N^{\boldsymbol{\varrho}}(\boldsymbol{\omega}, z) F^{\boldsymbol{\jmath}}(\boldsymbol{z} + \boldsymbol{\jmath}) = \mathbf{0} \tag{131}$$

By partial integrating, Eq. (131) becomes

$$N\_{\mathcal{Y}}^d(\mathbf{x}, \boldsymbol{\mathcal{y}}) - N^\sigma(\mathbf{x}, \boldsymbol{\mathfrak{x}}) F(\mathbf{x} + \boldsymbol{\mathfrak{y}}) - \int\_{\boldsymbol{\mathfrak{x}}}^{\boldsymbol{\mathfrak{o}}} \mathrm{d} \mathbf{z} N\_{\boldsymbol{\mathfrak{x}}}^\sigma(\mathbf{x}, \boldsymbol{\mathfrak{z}}) F(\mathbf{z} + \boldsymbol{\mathfrak{y}}) = \mathbf{0} \tag{132}$$

Use is made of the fact that *F*<sup>0</sup> ð Þ *z* þ *y* = *Fy*ð Þ¼ *z* þ *y Fz*ð Þ *z* þ *y* in (131). Making a weighing summation as follows:

$$\{l.h.s.\quad\text{of}\ (20)\} + \sigma\_3\{l.h.s.\quad\text{of}\ (22)\}\sigma\_3 - U(\text{x})\cdot\{l.h.s.\quad\text{of}\ (19)\} = 0$$

We find

$$\begin{aligned} \mathcal{N}\_x^d(\mathbf{x}, \boldsymbol{y}) + \sigma\_3 \mathcal{N}\_y^d(\mathbf{x}, \boldsymbol{y}) \sigma\_3 - U(\mathbf{x}) \mathcal{N}^p(\mathbf{x}, \boldsymbol{y}) - \mathcal{N}^p(\mathbf{x}, \mathbf{x}) F(\mathbf{x} + \boldsymbol{y}) - \sigma\_3 \mathcal{N}^p(\mathbf{x}, \mathbf{x}) F(\mathbf{x} + \boldsymbol{y}) \sigma\_3 - U(\mathbf{x}) F(\mathbf{x} + \boldsymbol{y}) \\ + \int\_{\mathbf{x}}^{\mathbf{x}} \mathop{\mathrm{d}\mathbf{x}} \Big{\mathrm{d}\mathbf{x}} \Big{\mathrm{d}\mathbf{x}} \Big{\mathrm{d}\mathbf{x}} \Big{(} N\_x^o(\mathbf{x}, \boldsymbol{z}) F(\mathbf{z} + \boldsymbol{y}) - \sigma\_3 N\_x^p(\mathbf{x}, \mathbf{z}) F(\mathbf{x} + \boldsymbol{y}) \mathcal{N}^d(\mathbf{x}, \mathbf{z}) G(\mathbf{z} + \boldsymbol{y}) \Big{)} = \mathbf{0} \end{aligned} \tag{133}$$

Since *F x*ð Þ is off-diagonal, *F x*ð Þ*σ*<sup>3</sup> ¼ �*σ*3*F x*ð Þ. Thus the terms involving with *F x*ð Þ <sup>þ</sup> *<sup>y</sup>* outside of integral are equal to �*iU x*ð Þ*N<sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>x</sup> <sup>σ</sup>*3*F x*ð Þ <sup>þ</sup> *<sup>y</sup>* by use of Eq. (123). Then (133) can be rewritten as

$$\begin{split} &A(\mathbf{x},\mathbf{y}) - iU(\mathbf{x})\mathcal{N}^{d}(\mathbf{x},\mathbf{z})\sigma\_{3}F(\mathbf{x}+\mathbf{y}) \\ &+ \int\_{\mathbf{x}}^{\infty} \text{d}z \left[ B(\mathbf{x},\mathbf{z})F(\mathbf{z}+\mathbf{y}) - iU(\mathbf{x})\mathcal{N}\_{x}^{d}(\mathbf{x},\mathbf{z})\sigma\_{3}F(\mathbf{z}+\mathbf{y}) - U(\mathbf{x})\mathcal{N}^{d}(\mathbf{x},\mathbf{z})G(\mathbf{z}+\mathbf{y}) \right] = \mathbf{0} \end{split} \tag{134}$$

If we choose

$$G(z+\jmath) = i\sigma\_3 F'(z+\jmath),\tag{135}$$

then

$$i\mathcal{U}(\mathbf{x})\mathcal{N}^d(\mathbf{x},\mathbf{x})\sigma\_3\mathcal{F}(\mathbf{x}+\mathbf{y}) + \int\_{\mathbf{z}}^{\infty} \text{d}\mathbf{z} \{i\mathcal{U}(\mathbf{x})\mathcal{N}^d\_{\mathbf{z}}(\mathbf{x},\mathbf{z})\sigma\_3\mathcal{F}(\mathbf{z}+\mathbf{y}) + \mathcal{U}(\mathbf{x})\mathcal{N}^d(\mathbf{x},\mathbf{z})\mathrm{i}\sigma\_3\mathcal{F}(\mathbf{z}+\mathbf{y})\} = \mathbf{0} \tag{136}$$

Thus, Eq. (134) becomes

$$A(\mathbf{x}, \boldsymbol{\uprho}) + \int\_{\mathbf{x}}^{\infty} \mathbf{d}z B(\mathbf{x}, z) F(\mathbf{z} + \boldsymbol{\uprho}) = \mathbf{0} \tag{137}$$

owing to the symmetry properties of *N<sup>o</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

*f x*ð Þ þ *y* 0

0 �*f x*ð Þ þ *y*

!

*F x*ð Þ¼ þ *y*

parameter *λ*<sup>2</sup>

as *x* ! �∞.

**formalism**

(128) and (129) can only has off-diagonal elements, we write

, in the reflectionless case, we choose

where *Cn*ð Þ*<sup>t</sup>* contains a time-dependent factor *ei*4*λ*<sup>4</sup>

be introduced in its solution, that is, *Cn* <sup>¼</sup> *<sup>e</sup><sup>β</sup>n*þ*iα<sup>n</sup> <sup>e</sup><sup>i</sup>*4*λ*<sup>4</sup>

convergence of the partial integral, we must let limx!<sup>∞</sup>*e<sup>i</sup>λ*<sup>2</sup>

tion (128) and (129), (144), and (145) for DNLSE with VBC.

equation with VBC in the reflectionless case. We can assume that

*N*

*n*¼1

*nx*, *hn*ð Þ� *<sup>y</sup> <sup>e</sup><sup>i</sup>λ*<sup>2</sup>

*f x*ð Þ¼ <sup>þ</sup> *<sup>y</sup> <sup>F</sup>*21ð Þ¼ *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>* <sup>X</sup>

*G x*ð Þ� , *<sup>t</sup> <sup>g</sup>*1ð Þ *<sup>x</sup>*, *<sup>t</sup>* , *<sup>g</sup>*2ð Þ *<sup>x</sup>*, *<sup>t</sup>* , <sup>⋯</sup>, *gN*ð Þ *<sup>x</sup>*, *<sup>t</sup>* � �

where *gn*ð Þ� *<sup>x</sup>*, *<sup>t</sup> Cn*ð Þ*<sup>t</sup> <sup>e</sup><sup>i</sup>λ*<sup>2</sup>

other hand, we assume that

**43**

upper half part of the complex plane of *λ*<sup>2</sup>

*f x*ð Þ¼ <sup>þ</sup> *<sup>y</sup>* <sup>X</sup>

; *G z*ð Þ¼ þ *y*

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

Considering the dependence of the Jost solutions on the squared spectral

*N*

*n*¼1

As is well known, Lax equations are linear equation so that a constant factor can

the center of soliton and *α<sup>n</sup>* expresses the initial phase up to a constant factor. Thus, the time-independent part of *Cn* is inessential and can be absorbed or normalized only by redefinition of the soliton center and initial phase. On the other hand, notice the terms generated by partial integral in (133)–(142), in order to ensure the

the *N* zero points of *a*ð Þ*λ* in the first quadrant of complex plane of *λ* (also in the

⋯⋯*λN*, although �*λn*, ð Þ *n* ¼ 12, ⋯, *N* in the third quadrant of the complex plane of *λ* are also the zero points of *a*ð Þ*λ* due to symmetry of Lax operator and transition matrix. Then Eq. (145) corresponds to the *N*-soliton solution in the reflectionless case, and we have completed the derivation and manifestation of Marchenko equa-

**3.4 A multi-soliton solution of the DNLS equation based upon pure Marchenko**

When there are *N* simple poles *λ*1, *λ*2, ⋯, *λ<sup>N</sup>* in the first quadrant of the complex plane of *λ*, the Marchenko equation will give a *N*-solition solution to the DNLS

*ny*

Here and hereafter the superscript T represents transposing of a matrix. On the

*<sup>N</sup>*11ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>N</sup>*11ð Þ *<sup>x</sup> H y*ð ÞT, *<sup>N</sup>*12ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>N</sup>*12ð Þ *<sup>x</sup> H y*ð Þ<sup>T</sup> (148)

by a standard procedure [29], due to a fact of the Lax operator *<sup>M</sup>* ! �*i*2*λ*<sup>4</sup>*σ*<sup>3</sup>

*Cn*ð Þ*t e iλ*2

ð Þ *<sup>x</sup>*, *<sup>y</sup>* and *<sup>N</sup><sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* , the function *f x*ð Þ <sup>þ</sup> *<sup>y</sup>* in

*<sup>n</sup>*ð Þ *<sup>x</sup>*þ*<sup>y</sup>* (145)

, which can be introduced

. It means that *β<sup>n</sup>* is related to

*<sup>n</sup><sup>x</sup>* <sup>¼</sup> 0, so we only consider

), that is, the discrete spectrum for *λ*1, *λ*2,

*gn*ð Þ *<sup>x</sup>*, *<sup>t</sup> hn*ð Þ� *<sup>y</sup> G x*ð Þ , *<sup>t</sup> H y*ð Þ<sup>T</sup> (146)

<sup>T</sup> (147)

, *n* ¼ 1, 2, ⋯, *N*, and

; *H y*ð Þ<sup>T</sup> � ð Þ *<sup>h</sup>*1ð Þ*<sup>y</sup>* , *<sup>h</sup>*2ð Þ*<sup>y</sup>* , <sup>⋯</sup>, *hN*ð Þ*<sup>y</sup>*

¼ *iσ*3*F*<sup>0</sup>

ð Þ *z* þ *y*

(144)

0 �*h z*ð Þ þ *y*

!

*h z*ð Þ þ *y* 0

*nt*

*nt*

Now substituting (135) into (129), we find

$$N^{o}(\mathbf{x},\boldsymbol{\uprho}) + F(\mathbf{x}+\boldsymbol{\uprho}) + \int\_{\boldsymbol{\uprho}}^{\boldsymbol{\uprho}} \mathrm{d}\mathbf{z} \mathbf{N}^{d}(\mathbf{x},\boldsymbol{\uprho}) i\sigma\_{3} F(\mathbf{z}+\boldsymbol{\uprho}) = \mathbf{0} \tag{138}$$

Making partial derivation with respect to *x* and *y*, respectively, on the l.h.s. of Eq. (138), we have

$$N\_x^{\rho}(\mathbf{x}, \boldsymbol{y}) + F'(\mathbf{x} + \boldsymbol{y}) - N^d(\mathbf{x}, \boldsymbol{x}) i\sigma\_3 F'(\mathbf{x} + \boldsymbol{y}) + \int\_x^{\infty} \mathrm{d}z N\_x^d(\mathbf{x}, \boldsymbol{z}) i\sigma\_3 F'(\mathbf{z} + \boldsymbol{y}) = \mathbf{0} \tag{139}$$

$$N\_{\mathcal{Y}}^{\boldsymbol{\varrho}}(\boldsymbol{x},\boldsymbol{y}) + F'(\boldsymbol{x}+\boldsymbol{y}) + \int\_{\boldsymbol{x}}^{\infty} \mathrm{d}\mathbf{z} \mathbf{N}^{d}(\boldsymbol{x},\boldsymbol{z}) i\sigma\_{3} F''(\boldsymbol{z}+\boldsymbol{y}) = \mathbf{0} \tag{140}$$

or

$$N\_y^o(\mathbf{x}, \boldsymbol{y}) + F'(\mathbf{x} + \boldsymbol{y}) - N^d(\mathbf{x}, \boldsymbol{x}) i\sigma\_3 F'(\mathbf{x} + \boldsymbol{y}) - \int\_{\mathbf{x}}^{\infty} \mathrm{d}z N\_x^d(\mathbf{x}, \boldsymbol{x}) i\sigma\_3 F'(\mathbf{z} + \boldsymbol{y}) = \mathbf{0} \tag{141}$$

Now we make a weighing summation as

$$\{l.h.s.\text{ of (29)}\} + \sigma\_3\{l.h.s.\text{ of (31)}\}\sigma\_3 + iU(\text{x})\{l.h.s.\text{ of (21)}\}\sigma\_3 = 0$$

Hence, we have

$$\begin{aligned} N\_{\mathbf{x}}^{\rho}(\mathbf{x},\mathbf{y}) + \sigma\_{3}N\_{\mathbf{y}}^{\rho}(\mathbf{x},\mathbf{y})\sigma\_{3} + iU(\mathbf{x})N\_{\mathbf{y}}^{d}(\mathbf{x},\mathbf{y})\sigma\_{3} \\ + F(\mathbf{x}+\mathbf{y}) + \sigma\_{3}F(\mathbf{x}+\mathbf{y})\sigma\_{3} - N^{d}(\mathbf{x},\mathbf{x})i\sigma\_{3}F'(\mathbf{x}+\mathbf{y}) - \sigma\_{3}N^{d}(\mathbf{x},\mathbf{x})i\sigma\_{3}F'(\mathbf{x}+\mathbf{y})\sigma\_{3} \\ + \int\_{\mathbf{x}}^{\infty} \mathbf{d} \mathbf{z} \left\{ N\_{\mathbf{x}}^{d}(\mathbf{x},\mathbf{z})i\sigma\_{3}F'(\mathbf{z}+\mathbf{y}) - \sigma\_{3}N\_{\mathbf{x}}^{d}(\mathbf{x},\mathbf{z})i\sigma\_{3}F'(\mathbf{z}+\mathbf{y})\sigma\_{3} + iU(\mathbf{x})N\_{\mathbf{x}}^{p}(\mathbf{x},\mathbf{z})F'(\mathbf{z}+\mathbf{y})\sigma\_{3} \right\} = 0 \end{aligned} \tag{142}$$

Noticing *F x*ð Þ*σ*<sup>3</sup> ¼ �*σ*3*F x*ð Þ, Eq. (142) becomes

$$B(\mathbf{x}, \boldsymbol{\eta}) + \int\_{\mathbf{x}}^{\infty} \mathbf{d}z A(\mathbf{x}, z) i\sigma\_3 F'(\mathbf{z} + \boldsymbol{\eta}) = \mathbf{0} \tag{143}$$

We find that, as long as we choose a suitable form for *G x*ð Þ þ *y* as well as *F x*ð Þ þ *y* according to Eq. (135), Eq. (128) and (129) will just satisfy the two conditions (126) and (127) derived from the first Lax Eq. (111). On the other hand, owing to the symmetry properties of *N<sup>o</sup>* ð Þ *<sup>x</sup>*, *<sup>y</sup>* and *<sup>N</sup><sup>d</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* , the function *f x*ð Þ <sup>þ</sup> *<sup>y</sup>* in (128) and (129) can only has off-diagonal elements, we write

$$F(\mathbf{x} + \mathbf{y}) = \begin{pmatrix} \mathbf{0} & \overline{-f(\mathbf{x} + \mathbf{y})} \\ f(\mathbf{x} + \mathbf{y}) & \mathbf{0} \end{pmatrix};\\ G(\mathbf{z} + \mathbf{y}) = \begin{pmatrix} \mathbf{0} & -\overline{h(\mathbf{z} + \mathbf{y})} \\ h(\mathbf{z} + \mathbf{y}) & \mathbf{0} \end{pmatrix} = i\sigma\_3 F(\mathbf{z} + \mathbf{y}) \tag{144}$$

Considering the dependence of the Jost solutions on the squared spectral parameter *λ*<sup>2</sup> , in the reflectionless case, we choose

$$f(\mathbf{x} + \mathbf{y}) = \sum\_{n=1}^{N} \mathbf{C}\_{n}(t) e^{j\lambda\_{n}^{2}(\mathbf{x} + \mathbf{y})} \tag{145}$$

where *Cn*ð Þ*<sup>t</sup>* contains a time-dependent factor *ei*4*λ*<sup>4</sup> *nt* , which can be introduced by a standard procedure [29], due to a fact of the Lax operator *<sup>M</sup>* ! �*i*2*λ*<sup>4</sup>*σ*<sup>3</sup> as *x* ! �∞.

As is well known, Lax equations are linear equation so that a constant factor can be introduced in its solution, that is, *Cn* <sup>¼</sup> *<sup>e</sup><sup>β</sup>n*þ*iα<sup>n</sup> <sup>e</sup><sup>i</sup>*4*λ*<sup>4</sup> *nt* . It means that *β<sup>n</sup>* is related to the center of soliton and *α<sup>n</sup>* expresses the initial phase up to a constant factor. Thus, the time-independent part of *Cn* is inessential and can be absorbed or normalized only by redefinition of the soliton center and initial phase. On the other hand, notice the terms generated by partial integral in (133)–(142), in order to ensure the convergence of the partial integral, we must let limx!<sup>∞</sup>*e<sup>i</sup>λ*<sup>2</sup> *<sup>n</sup><sup>x</sup>* <sup>¼</sup> 0, so we only consider the *N* zero points of *a*ð Þ*λ* in the first quadrant of complex plane of *λ* (also in the upper half part of the complex plane of *λ*<sup>2</sup> ), that is, the discrete spectrum for *λ*1, *λ*2, ⋯⋯*λN*, although �*λn*, ð Þ *n* ¼ 12, ⋯, *N* in the third quadrant of the complex plane of *λ* are also the zero points of *a*ð Þ*λ* due to symmetry of Lax operator and transition matrix. Then Eq. (145) corresponds to the *N*-soliton solution in the reflectionless case, and we have completed the derivation and manifestation of Marchenko equation (128) and (129), (144), and (145) for DNLSE with VBC.

## **3.4 A multi-soliton solution of the DNLS equation based upon pure Marchenko formalism**

When there are *N* simple poles *λ*1, *λ*2, ⋯, *λ<sup>N</sup>* in the first quadrant of the complex plane of *λ*, the Marchenko equation will give a *N*-solition solution to the DNLS equation with VBC in the reflectionless case. We can assume that

$$f(\mathbf{x} + \mathbf{y}) = F\_{21}(\mathbf{x} + \mathbf{y}) = \sum\_{n=1}^{N} \mathbf{g}\_n(\mathbf{x}, t) h\_n(\mathbf{y}) \equiv \mathbf{G}(\mathbf{x}, t) H(\mathbf{y})^T \tag{146}$$

where *gn*ð Þ� *<sup>x</sup>*, *<sup>t</sup> Cn*ð Þ*<sup>t</sup> <sup>e</sup><sup>i</sup>λ*<sup>2</sup> *nx*, *hn*ð Þ� *<sup>y</sup> <sup>e</sup><sup>i</sup>λ*<sup>2</sup> *ny* , *n* ¼ 1, 2, ⋯, *N*, and

$$\mathbf{G}(\mathbf{x},t) \equiv \left( \mathbf{g}\_1(\mathbf{x},t), \mathbf{g}\_2(\mathbf{x},t), \dots, \mathbf{g}\_N(\mathbf{x},t) \right)\_\circ H(\mathbf{y})^\mathsf{T} \equiv \left( h\_1(\mathbf{y}), h\_2(\mathbf{y}), \dots, h\_N(\mathbf{y}) \right)^\mathsf{T} \tag{147}$$

Here and hereafter the superscript T represents transposing of a matrix. On the other hand, we assume that

$$N\_{11}(\mathbf{x}, \mathbf{y}) = N\_{11}(\mathbf{x}) H(\mathbf{y})^{\mathrm{T}}, \\ N\_{12}(\mathbf{x}, \mathbf{y}) = N\_{12}(\mathbf{x}) \overline{H}(\mathbf{y})^{\mathrm{T}} \tag{148}$$

Then

$$\begin{split} F\_{12}(\mathbf{x} + \mathbf{y}) &= -\overline{F}\_{12}(\mathbf{x} + \mathbf{y}) = -\overline{G}(\mathbf{x})\overline{H}(\mathbf{y})^{\mathrm{T}}, \; F\_{12}'(\mathbf{x} + \mathbf{y}) = i\overline{i\_n^2}\overline{C}\_n e^{-i\mathbf{i}\_n^2(\mathbf{x} + \mathbf{y})} \\ &= -\overline{G'(\mathbf{x})H}(\mathbf{y})^{\mathrm{T}} \end{split} \tag{149}$$

Substituting (146)–(149) into the Marchenko equation (128) and (129), we have

$$\begin{cases} N\_{11}(\mathbf{x})\mathbf{H}(\mathbf{y})^{\mathsf{T}} + N\_{12}(\mathbf{x})\int\_{\mathbf{x}}^{\mathbf{s}\mathsf{s}} \mathrm{d}\mathbf{z}\overline{\mathbf{H}}(\mathbf{z})^{\mathsf{T}}\mathbf{G}(\mathbf{z})\mathbf{H}(\mathbf{y})^{\mathsf{T}} = \mathbf{0} \\ N\_{12}(\mathbf{x})\overline{\mathbf{H}}(\mathbf{y})^{\mathsf{T}} - \overline{\mathbf{G}}(\mathbf{x})\overline{\mathbf{H}}(\mathbf{y})^{\mathsf{T}} - i\mathcal{N}\_{11}(\mathbf{x})\int\_{\mathbf{x}}^{\mathbf{s}\mathsf{s}} \mathrm{d}\mathbf{z}\mathbf{H}(\mathbf{z})^{\mathsf{T}}\overline{\mathbf{G}}'(\mathbf{z})\overline{\mathbf{H}}(\mathbf{y})^{\mathsf{T}} = \mathbf{0} \end{cases} \tag{150}$$

or

$$N\_{11}(\mathbf{x}) + N\_{12}(\mathbf{x})\Delta\_1(\mathbf{x}) = \mathbf{0},\tag{151}$$

Substituting (160) and (161) into Eq. (124), we thus attain the *N*-soliton solution

G

G

� � <sup>¼</sup> det Ið Þ� <sup>þ</sup> *<sup>i</sup>*Δ1Δ<sup>2</sup> *<sup>D</sup>* (164)

� � <sup>Δ</sup><sup>2</sup> <sup>þ</sup> *<sup>H</sup>*<sup>T</sup>

*gm*, Mð Þ<sup>2</sup> *mn* � <sup>Δ</sup><sup>2</sup> <sup>þ</sup> *<sup>H</sup>*<sup>T</sup>

The complex constant factor *cn*<sup>0</sup> can be absorbed into the soliton center and

� �*<sup>t</sup>* � � <sup>¼</sup> <sup>4</sup>*κn*ð Þ *<sup>x</sup>* � *xn*<sup>0</sup> � *<sup>υ</sup>nt* ; *<sup>κ</sup><sup>n</sup>* <sup>¼</sup> <sup>4</sup>*μnνn*; *<sup>υ</sup>*<sup>n</sup>

*N*

*r*¼1

where M1 *n*1, *n*2, ⋯, *nr* ð Þ ; *m*1, *m*2, ⋯*mr* denotes a minor, which is the determinant of a submatrix of M1, consisting of elements belonging to not only (*n*1, *n*2, … , *nr*)

M1 *n*1, *n*2, ⋯, *nr* ð Þ ; *m*1, *m*2, ⋯*mr* M2 *m*1, *m*2, ⋯, *mr* ð Þ , *n*1, *n*2, ⋯, *nr*

*nx* <sup>¼</sup> *cnoe*

*i*4*λ*<sup>4</sup> *nt e i*2*λ*<sup>2</sup> *nx* � *<sup>e</sup>*

*<sup>n</sup>* � *<sup>v</sup>*<sup>2</sup> *n* � �<sup>2</sup> � <sup>16</sup>*μ*<sup>2</sup>

*<sup>i</sup>αno* ; *<sup>n</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>N</sup>* (168)

X 1≤*n*<sup>1</sup> <*n*<sup>2</sup> < ⋯ <*nr* ≤ *N*

*i*2*λ*<sup>2</sup>

� � � det Ið Þ <sup>þ</sup> *<sup>i</sup>*Δ1Δ<sup>2</sup> , (163)

*G* h i � � <sup>≔</sup>detð Þ *<sup>I</sup>* <sup>þ</sup> M1M2

> *G* � �

> > �*θ<sup>n</sup> e*

*nv*2 *n* h i•*<sup>t</sup>* <sup>þ</sup> *<sup>α</sup>no*;*cn*<sup>0</sup>

*mn* <sup>¼</sup> *hm*

det I � *i*Δ1Δ<sup>2</sup> � � det Ið Þ þ *i*Δ1Δ<sup>2</sup>

¼ �2*CD=D*<sup>2</sup>

(162)

(165)

(166)

(169)

*λ*2 *m λ*2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n gn*

*<sup>i</sup>φ<sup>n</sup>* (167)

X 1≤ *m*<sup>1</sup> < *m*<sup>2</sup> < ⋯ < *mr* ≤ *N*

� � �

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

G

Firstly, we can prove identity (164) by means of Binet-Cauchy formula.

By means of some linear algebraic techniques, especially the Binet-Cauchy formula for some special matrices (see the Appendices 2–3 in Part2), the determinant *D* and *C* can be expanded explicitly as a summation of all possible principal minors.

as follows in a pure Marchenko formalism.

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

and we will prove that in (136)

det *<sup>I</sup>* � *<sup>i</sup>*Δ1Δ<sup>2</sup> � *<sup>i</sup>*Δ1*H*<sup>T</sup>

� �

*mn* ¼ *hn*

*gn*ð Þ *x*, *t hn*ð Þ¼ *x Cn*ð Þ*t e*

*<sup>n</sup>* � *<sup>v</sup>*<sup>2</sup> *n*

*G* � � <sup>¼</sup> <sup>1</sup> <sup>þ</sup><sup>X</sup>

*<sup>n</sup>* � *<sup>v</sup>*<sup>2</sup> *n* � �*<sup>x</sup>* <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

det I <sup>þ</sup> *<sup>i</sup>*ΔiΔ<sup>2</sup> <sup>þ</sup> <sup>H</sup><sup>T</sup>

*<sup>D</sup>* � det Ið Þ <sup>þ</sup> *<sup>i</sup>*Δ1Δ<sup>2</sup> ,*<sup>C</sup>* � det I <sup>þ</sup> *<sup>i</sup>*Δ1Δ<sup>2</sup> <sup>þ</sup> HT

det I � *<sup>i</sup>*Δ1Δ<sup>2</sup> � *<sup>i</sup>*Δ1H<sup>T</sup>

*G* � � <sup>¼</sup> det *<sup>I</sup>* þ �*i*Δ<sup>1</sup>

> 1 *λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *m*

G � � � det Ið Þ <sup>þ</sup> *<sup>i</sup>*Δ1Δ<sup>2</sup>

det I � *<sup>i</sup>*Δ1Δ<sup>2</sup> � *<sup>i</sup>*Δ1H<sup>T</sup>

*uN*ð Þ¼� *x*, *t* 2

where

where

ð Þ Μ<sup>1</sup> *nm* � �*i*Δ<sup>1</sup>

initial phase by redefining

here *λ<sup>n</sup>* ¼ *μ<sup>n</sup>* þ *ivn*, and

*<sup>θ</sup><sup>n</sup>* � <sup>4</sup>*μnvn <sup>x</sup>* � *xn*<sup>0</sup> <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

*<sup>n</sup>* � *<sup>v</sup>*<sup>2</sup> *n* � �; *<sup>φ</sup><sup>n</sup>* � <sup>2</sup> *<sup>μ</sup>*<sup>2</sup>

det *<sup>I</sup>* � *<sup>i</sup>*Δ1Δ<sup>2</sup> � *<sup>i</sup>*Δ1*H*<sup>T</sup>

rows but also columns (*m*1, *m*2, … , *mr*).

¼ �<sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

� *e* <sup>4</sup>*κnxno e*

**45**

$$i N\_{12}(\mathbf{x}) - i N\_{11}(\mathbf{x}) \Delta\_2(\mathbf{x}) = \overline{G}(\mathbf{x}) \tag{152}$$

here

$$\Delta\_1(\mathbf{x}) = \int\_{\mathbf{x}}^{\infty} \overline{\mathbf{H}}(\mathbf{z})^T \mathbf{G}(\mathbf{z}) \, \mathbf{d}z,\\ \Delta\_2(\mathbf{x}) = \int\_{\mathbf{x}}^{\infty} \mathbf{H}(\mathbf{z})^T \overline{\mathbf{G}}'(\mathbf{z}) \, \mathbf{d}z \tag{153}$$

Both of them are *N* � *N* matrices and their matrix element are, respectively, expressed as

$$\left(\Delta\_1(\mathbf{x})\_{mn} = \int\_{\varkappa}^{\infty} e^{-i\overline{\lambda\_m^2 x}} \mathbf{C}\_n e^{i\lambda\_n^2 x} \,\mathrm{d}z = \overline{h}\_m(\infty) \frac{-i}{\overline{\lambda\_m^2} - \lambda\_n^2} \mathbf{g}\_n(\infty) \tag{154}$$

$$\Delta\_2(\mathbf{x})\_{mn} = \int\_{\infty}^{\infty} e^{-i\lambda\_m^2 x} \overline{\mathbf{C}\_n} \left(-i\overline{\lambda\_n^2}\right) e^{-i\overline{\lambda\_n^2}x} d\mathbf{z} = h\_m(\mathbf{x}) \frac{\overline{\lambda\_n^2}}{\lambda\_m^2 - \overline{\lambda\_n^2}} \overline{\mathbf{g}}\_n(\mathbf{x}) \tag{155}$$

From (151) and (152), we immediately get

$$N\_{11}(\mathbf{x}) = -\overline{\mathbf{G}}(\mathbf{x})[\mathbf{1} + i\Delta\_1(\mathbf{x})\Delta\_2(\mathbf{x})]^{-1}\Delta\_1(\mathbf{x})\tag{156}$$

$$N\_{12}(\mathbf{x}) = \overline{\mathbf{G}}(\mathbf{x}) \left[\mathbf{1} + i\Delta\_1(\mathbf{x})\Delta\_2(\mathbf{x})\right]^{-1} \tag{157}$$

from (148), (156), and (157), we have

$$N\_{11}(\mathbf{x}, \mathbf{y}) = -\overline{G}(\mathbf{x}) [\mathbf{I} + i\Delta\_1(\mathbf{x})\Delta\_2(\mathbf{x})]^{-1} \Delta\_1(\mathbf{x}) H(\mathbf{y})^T \tag{158}$$

$$N\_{12}(\mathbf{x}, \mathbf{y}) = \overline{\mathbf{G}}(\mathbf{x}) [\mathbf{I} + i\Delta\_1(\mathbf{x})\Delta\_2(\mathbf{x})]^{-1} \overline{\mathbf{H}}(\mathbf{y})^\mathrm{T} \tag{159}$$

then

$$\begin{split} N\_{11}(\mathbf{x}, \mathbf{x}) &= i \text{Tr} \left\{ i \Delta\_{1}(\mathbf{x}) \mathbf{H}(\mathbf{x})^{\text{T}} \overline{\mathbf{G}}(\mathbf{x}) [\mathbf{I} + i \Delta\_{1}(\mathbf{x}) \Delta\_{2}(\mathbf{x})]^{-1} \right\} \\ &= i \left\{ \frac{\det \left[ \mathbf{I} + i \Delta\_{1}(\mathbf{x}) \Delta\_{2}(\mathbf{x}) + i \Delta\_{1}(\mathbf{x}) \mathbf{H}(\mathbf{x})^{\text{T}} \overline{\mathbf{G}}(\mathbf{x}) \right]}{\det[\mathbf{I} + i \Delta\_{1}(\mathbf{x}) \Delta\_{2}(\mathbf{x})]} - \mathbf{1} \right\} \end{split} \tag{160}$$

and

$$N\_{12}(\mathbf{x}, \mathbf{x}) = \text{Tr}\left\{ \overline{\mathbf{H}}(\mathbf{x})^{\mathsf{T}} \overline{\mathbf{G}}(\mathbf{x}) [\mathbf{I} + i\Delta\_{1}(\mathbf{x})\Delta\_{2}(\mathbf{x})]^{-1} \right\} = \frac{\text{det}\left[\mathbf{I} + i\Delta\_{1}(\mathbf{x})\Delta\_{2}(\mathbf{x}) + \overline{\mathbf{H}}^{\mathsf{T}}(\mathbf{x})\overline{\mathbf{G}}(\mathbf{x})\right]}{\text{det}[\mathbf{I} + i\Delta\_{1}(\mathbf{x})\Delta\_{2}(\mathbf{x})]} - \mathbf{1} \tag{161}$$

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

Substituting (160) and (161) into Eq. (124), we thus attain the *N*-soliton solution as follows in a pure Marchenko formalism.

$$u\_N(\mathbf{x}, t) = -2 \frac{\det\left(\mathbf{I} + i\Delta\_1 \Delta\_2 + \overline{\mathbf{H}}^T \overline{\mathbf{G}}\right) - \det(\mathbf{I} + i\Delta\_1 \Delta\_2)}{\det\left(\mathbf{I} - i\overline{\Delta}\_1 \overline{\Delta}\_2 - i\overline{\Delta}\_1 \overline{\mathbf{H}}^T \mathbf{G}\right)} \cdot \frac{\det(\mathbf{I} - i\overline{\Delta}\_1 \overline{\Delta}\_2)}{\det(\mathbf{I} + i\Delta\_1 \Delta\_2)} = -2C\overline{\mathcal{D}}/D^2 \tag{162}$$

where

$$D \equiv \det(\mathbf{I} + i\Delta\_1 \Delta\_2),\\ \mathbf{C} \equiv \det\left(\mathbf{I} + i\Delta\_1 \Delta\_2 + \overline{\mathbf{H}}^T \overline{\mathbf{G}}\right) - \det(\mathbf{I} + i\Delta\_1 \Delta\_2),\tag{163}$$

and we will prove that in (136)

$$\det\left(\mathbf{I} - i\overline{\Delta}\_1 \overline{\Delta}\_2 - i\overline{\Delta}\_1 \overline{\mathbf{H}}^T \mathbf{G}\right) = \det(\mathbf{I} + i\Delta\_1 \Delta\_2) \equiv D \tag{164}$$

By means of some linear algebraic techniques, especially the Binet-Cauchy formula for some special matrices (see the Appendices 2–3 in Part2), the determinant *D* and *C* can be expanded explicitly as a summation of all possible principal minors. Firstly, we can prove identity (164) by means of Binet-Cauchy formula.

$$\det\left(I - i\overline{\Delta}\_1 \overline{\Delta}\_2 - i\overline{\Delta}\_1 \overline{H}^T G\right) = \det\left[I + \left(-i\overline{\Delta}\_1\right)\left(\overline{\Delta}\_2 + \overline{H}^T G\right)\right] \coloneqq \det(I + \mathbf{M}\_1 \mathbf{M}\_2) \tag{165}$$

where

$$\mathbf{u}(\mathbf{M}\_1)\_{nm} \equiv \left(-i\overline{\boldsymbol{\Delta}}\_1\right)\_{mn} = h\_n \frac{\mathbf{1}}{\boldsymbol{\lambda}\_n^2 - \overline{\boldsymbol{\lambda}\_m^2}} \overline{\mathbf{g}}\_m,\\ (\mathbf{M}\_2)\_{mn} \equiv \left(\overline{\boldsymbol{\Delta}}\_2 + \overline{\boldsymbol{H}}^T \mathbf{G}\right)\_{mn} = \overline{h}\_m \frac{\overline{\boldsymbol{\lambda}\_m^2}}{\overline{\boldsymbol{\lambda}\_m^2} - \overline{\boldsymbol{\lambda}\_n^2}} \mathbf{g}\_n \tag{166}$$

The complex constant factor *cn*<sup>0</sup> can be absorbed into the soliton center and initial phase by redefining

$$\mathcal{g}\_n(\varkappa, t) h\_n(\varkappa) = \mathcal{C}\_n(t) e^{i2\lambda\_n^2 \varkappa} = \mathcal{c}\_{no} e^{i4\lambda\_n^4 t} e^{i2\lambda\_n^2 \varkappa} \equiv e^{-\theta\_n} e^{i\rho\_n} \tag{167}$$

here *λ<sup>n</sup>* ¼ *μ<sup>n</sup>* þ *ivn*, and

$$\begin{split} \theta\_n &\equiv 4\mu\_n v\_n \left[ \mathbf{x} - \mathbf{x}\_{n0} + 4 \left( \mu\_n^2 - v\_n^2 \right) t \right] = 4\kappa\_n (\mathbf{x} - \mathbf{x}\_{n0} - \nu\_n t); \mathbf{x}\_n = 4\mu\_n \nu\_n; v\_n \\ &= -4 \left( \mu\_n^2 - v\_n^2 \right); \ \mu\_n \equiv 2 \left( \mu\_n^2 - v\_n^2 \right) \mathbf{x} + \left[ 4 \left( \mu\_n^2 - v\_n^2 \right)^2 - 16 \mu\_n^2 v\_n^2 \right] \mathbf{\hat{r}} + a\_{no}; c\_{n0} \\ &\equiv e^{4\kappa\_n \chi\_{\mathbf{z}}} e^{i\alpha\_{\mathbf{z}}}; n = 1, 2, \cdots, N \\ &\quad \rightarrow \quad \rightarrow \quad N \end{split} \tag{168}$$

$$\det\left(I - i\overline{\Delta}\_1 \overline{\Delta}\_2 - i\overline{\Delta}\_1 \overline{H}^T G\right) = 1 + \sum\_{r=1}^{\cdot \cdot \cdot} \sum\_{1 \le n\_1 < n\_2 < \cdots < n\_r \le N} \sum\_{\substack{1 \le m\_1 < m\_2 < \cdots < m\_r \le N \\ 1 \le n\_1 < m\_2 < \cdots < m\_r \le N}} \tag{16}$$
  $\mathbf{M}\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r)$  $\mathbf{M}\_2(m\_1, m\_2, \cdots, m\_r, n\_1, n\_2, \cdots, n\_r)$ 

where M1 *n*1, *n*2, ⋯, *nr* ð Þ ; *m*1, *m*2, ⋯*mr* denotes a minor, which is the determinant of a submatrix of M1, consisting of elements belonging to not only (*n*1, *n*2, … , *nr*) rows but also columns (*m*1, *m*2, … , *mr*).

*Nonlinear Optics - From Solitons to Similaritons*

$$\mathbf{M}\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots m\_r) = \prod\_{n, m} \frac{h\_n \overline{\mathbf{g}}\_m}{\lambda\_n^2 - \overline{\lambda\_m^2}} \prod\_{n < n', m < m'} \left(\lambda\_n^2 - \lambda\_{n'}^2\right) \overline{\left(\overline{\lambda\_{m'}^2} - \overline{\lambda\_m^2}\right)} \tag{170}$$

$$\mathbf{M}\_2(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots n\_r) = \prod\_{n,m} \frac{\overline{h}\_m \mathbf{g}\_n \overline{\boldsymbol{\lambda}\_m^2}}{\overline{\boldsymbol{\lambda}\_m^2} - \boldsymbol{\lambda}\_n^2} \prod\_{n$$

where *n*, *n*<sup>0</sup> ∈ f g *n*1, *n*2, ⋯, *nr* , *m*, *m*<sup>0</sup> ∈f g *m*1, *m*2, ⋯, *mr* , then M1 *n*1, *n*2, ⋯, *nr* ð Þ ; *m*1, *m*2, ⋯*mr* M2 *m*1, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯*nr*

$$=(-1)^{r} \prod\_{n,m} \frac{e^{-\theta\_{n}} e^{i\phi\_{n}} e^{-\theta\_{m}} e^{-i\rho\_{m}} \overline{\lambda\_{m}^{2}}}{\left(\overline{\lambda\_{n}^{2}} - \overline{\lambda\_{m}^{2}}\right)^{2}} \prod\_{n$$

If we define matrices *Q*<sup>1</sup> ¼ *i*Δ<sup>1</sup> and *Q*<sup>2</sup> ¼ Δ2, then we can similarly attain

$$D = \det(\mathbf{I} + i\Delta\_1 \Delta\_2) = \det(\mathbf{I} + \mathbf{Q}\_1 \mathbf{Q}\_2) = \mathbf{1} + \sum\_{r=1}^{N} \sum\_{1 \le n\_1 < n\_2 < \cdots < n\_r \le N} \sum\_{1 \le m\_1 < m\_2 < \cdots < m\_r \le N} $$
  $Q\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r)$  $Q\_2(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r)$ 

and

$$\mathbf{Q}\_{1}(n\_{1},n\_{2},\cdots,n\_{r};m\_{1},m\_{2},\cdots m\_{r})\mathbf{Q}\_{2}(m\_{1},m\_{2},\cdots,m\_{r};n\_{1},n\_{2},\cdots,n\_{r})$$

$$=(-1)^{r}\prod\_{n,m}\frac{e^{-\theta\_{n}}\boldsymbol{e}^{i\boldsymbol{\rho}\_{m}}e^{-\theta\_{n}}\boldsymbol{e}^{-i\boldsymbol{\rho}\_{n}}\overline{\boldsymbol{\lambda}\_{n}^{2}}}{\left(\boldsymbol{\lambda}\_{m}^{2}-\overline{\boldsymbol{\lambda}\_{n}^{2}}\right)^{2}}\prod\_{n$$

(173)

detð Þ¼ *<sup>I</sup>* <sup>þ</sup> <sup>Ω</sup>1Ω<sup>2</sup> <sup>1</sup> <sup>þ</sup><sup>X</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

(with *m*<sup>1</sup> ¼ 0, *m*<sup>2</sup> ≥1). Due to (175), we have

X 1 ≤ *m*<sup>2</sup> < *m*<sup>3</sup> < ⋯ < *mr* ≤ *N*

> *n hn* Y *m gm*

> *n gn* Y *m hm*

Ω<sup>1</sup> *n*1, *n*2, ⋯, *nr* ð Þ ; 0, *m*2, ⋯*mr* Ω<sup>2</sup> 0, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯, *nr*

**3.5 The special examples for one- and two-soliton solutions**

*<sup>D</sup>*<sup>1</sup> <sup>¼</sup> *<sup>Q</sup>*1ð Þ *<sup>n</sup>*<sup>1</sup> <sup>¼</sup> 1; *<sup>m</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> *<sup>Q</sup>*2ð*m*<sup>1</sup> <sup>¼</sup> 1; *<sup>n</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>Þ ¼ <sup>1</sup> � *<sup>g</sup>*1*h*1*g*1*h*1*λ*<sup>2</sup>

(173), (177), (174), and (180), we have

� �ð Þ<sup>1</sup> *<sup>r</sup>*þ<sup>1</sup>

Y *n*<*n*<sup>0</sup> , *m* < *m*<sup>0</sup>

Y *m λ*2 *m*,

> *λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � �<sup>2</sup>

here *n*, *n*<sup>0</sup> ∈ð Þ *n*1, *n*2, ⋯, *nr* , *m*, *m*<sup>0</sup> ∈ ð Þ *m*2, ⋯, *mr* in (178)–(180). Finally, substituting (174) into (173), (180) into (177), and (173 and 177) into (162), we thus attain the explicit *N*-soliton solution to the DNLS equation with VBC under the reflectionless case, based on a pure Marchenko formalism and in no need of the concrete spectrum expression of *a*ð Þ*λ* . Obviously, the *N*-soliton solution permits uncertain complex constants *cn*<sup>0</sup> ð Þ *n* ¼ 1, 2, ⋯, *N* as well as an arbitrary global constant phase factor.

In the case of one simple pole and one-soliton solution as *N* ¼ 1, according to

*<sup>C</sup>*<sup>1</sup> <sup>¼</sup> <sup>Ω</sup>1ð Þ *<sup>n</sup>*<sup>1</sup> <sup>¼</sup> 1; *<sup>m</sup>*<sup>1</sup> <sup>¼</sup> <sup>0</sup> <sup>Ω</sup>2ð*m*<sup>1</sup> <sup>¼</sup> 0; *<sup>n</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>Þ ¼ *<sup>g</sup>*1*h*1ð Þ �<sup>1</sup> <sup>1</sup>þ<sup>1</sup> <sup>¼</sup> *<sup>g</sup>*1*h*<sup>1</sup> (181)

From (167) and (168), we have (suppose *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>*1*e<sup>i</sup>β*<sup>1</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *iv*<sup>1</sup> and

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>g</sup>*1*h*<sup>1</sup>

1

� � �

�<sup>2</sup> *<sup>λ</sup>*<sup>2</sup> 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> (182)

*C* ¼ detð Þ� *I* þ Ω1Ω<sup>2</sup> detð Þ *I* þ *i*Δ1Δ<sup>2</sup>

<sup>Ω</sup><sup>1</sup> *<sup>n</sup>*1, *<sup>n</sup>*2, <sup>⋯</sup>, *nr* <sup>ð</sup> ; 0, *<sup>m</sup>*2, <sup>⋯</sup>*mr*Þ ¼ <sup>Y</sup>

<sup>Ω</sup><sup>2</sup> 0, *<sup>m</sup>*2, <sup>⋯</sup>, *mr* <sup>ð</sup> ; *<sup>n</sup>*1, *<sup>n</sup>*2, <sup>⋯</sup>*nr*Þ ¼ <sup>Y</sup>

X 1≤*n*<sup>1</sup> <*n*<sup>2</sup> < ⋯ <*nr* ≤ *N*

which leads to

Y *n e* �*θ<sup>n</sup> e* �*iφ<sup>n</sup>* Y *m e* �*θ<sup>m</sup> e iφ<sup>m</sup>*

¼ �ð Þ<sup>1</sup> *<sup>r</sup>*þ<sup>1</sup>

*<sup>c</sup>*<sup>10</sup> <sup>¼</sup> *<sup>e</sup>*<sup>4</sup>*κ*1*x*<sup>10</sup> *<sup>e</sup><sup>i</sup>α*<sup>10</sup> )

**47**

<sup>¼</sup> <sup>X</sup>*<sup>N</sup> r*¼1

*N*

X 1≤*n*<sup>1</sup> ≤*n*<sup>2</sup> < ⋯ <*nr* ≤ *N*

Ω<sup>1</sup> *n*1, *n*2, ⋯, *nr* ð Þ ; *m*1, *m*2, ⋯*mr* Ω<sup>2</sup> *m*1, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯, *nr*

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

The above summation obviously can be decomposed into two parts: one is extended to *m*<sup>1</sup> = 0 and the other extended to *m*<sup>1</sup> ≥ 1. Subtracted from (176), the part that is extended to *m*<sup>1</sup> ≥ 1, the remaining parts of (176) is just *C* in Eq. (163)

> Y *n*<*n*<sup>0</sup> , *m* < *m*<sup>0</sup>

> Y *n*<*n*<sup>0</sup> , *m* < *m*<sup>0</sup>

X 0≤ *m*<sup>1</sup> < *m*<sup>2</sup> < ⋯ < *mr* ≤ *N*

Ω<sup>1</sup> *n*1, *n*2, ⋯, *nr* ð Þ ; 0, *m*2, ⋯, *mr* Ω<sup>2</sup> 0, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯, *nr*

*<sup>m</sup>*<sup>0</sup> � *<sup>λ</sup>*<sup>2</sup> *m* � �Y

*<sup>m</sup>*<sup>0</sup> � *<sup>λ</sup>*<sup>2</sup> *m* � �Y

*n*, *m*

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � � *<sup>λ</sup>*<sup>2</sup>

*λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*0 � � *<sup>λ</sup>*<sup>2</sup>

*λ*2 *<sup>m</sup>*<sup>0</sup> � *<sup>λ</sup>*<sup>2</sup> *m* � �<sup>2</sup>Y (176)

(177)

1 *λ*2 *<sup>n</sup>* � *<sup>λ</sup>*<sup>2</sup> *m* (178)

1 *λ*2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n*

(179)

Y *m λ*2 *m*,

(180)

*n*, *m*

*n*, *m*

1

*λ*2 *<sup>m</sup>* � *<sup>λ</sup>*<sup>2</sup> *n* � �<sup>2</sup>

*r*¼1

where *n*, *n*<sup>0</sup> ∈ f g *n*1, *n*2, ⋯, *nr* , *m*, *m*<sup>0</sup> ∈ f g *m*1, *m*2, ⋯, *mr* . Comparing (172) and (174), we find the following permutation symmetry between them

$$\mathbf{M}\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r)\mathbf{M}\_2(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r)$$

$$=\mathbf{Q}\_1(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r)\mathbf{Q}\_2(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r)$$

Using above identity, comparing (169), (172), (173), and (174), we find that identity (164) holds and complete the computation of *D.*

Secondly, we compute the most complicate determinant *C* in (163). In order to calculate det *<sup>I</sup>* <sup>þ</sup> *<sup>i</sup>*Δ1Δ<sup>2</sup> <sup>þ</sup> *<sup>H</sup><sup>T</sup> G* � �, we introduce an *<sup>N</sup>* � ð Þ *<sup>N</sup>* <sup>þ</sup> <sup>1</sup> matrix <sup>Ω</sup><sup>1</sup> and an ð Þ� *N* þ 1 *N* matrix Ω<sup>2</sup>

$$\begin{aligned} \left(\begin{array}{c} \left(\boldsymbol{\Omega}\_{1}\right)\_{nm} = \left(i\boldsymbol{\Delta}\_{1}\right)\_{nm}, \left(\boldsymbol{\Omega}\_{1}\right)\_{n0} = \overline{h}\_{n} = \frac{\overline{h}\_{n}\overline{\lambda\_{n}^{2}}}{\overline{\lambda\_{n}^{2}} - \mathbf{0}^{2}}; \left(\boldsymbol{\Omega}\_{2}\right)\_{mn} = \left(\boldsymbol{\Delta}\_{2}\right)\_{mn}, \\\\ \left(\boldsymbol{\Omega}\_{2}\right)\_{0n} = \overline{g}\_{n} = \frac{-\overline{\lambda\_{n}^{2}}\overline{g}\_{n}}{\mathbf{0}^{2} - \overline{\lambda\_{n}^{2}}} \end{array} \end{aligned} \tag{175}$$

with *n*, *m* = 1, 2, … , *N*. We thus have

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

$$\det(I + \Omega\_1 \Omega\_2) = \mathbf{1} + \sum\_{r=1}^{N} \sum\_{1 \le n\_1 \le n\_2 < \cdots < n\_r \le N0} \sum\_{0 \le m\_1 < m\_2 < \cdots < m\_r \le N} \tag{176}$$
 
$$\Omega\_1(n\_1, n\_2, \cdots, n\_r; m\_1, m\_2, \cdots, m\_r) \Omega\_2(m\_1, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r)$$

The above summation obviously can be decomposed into two parts: one is extended to *m*<sup>1</sup> = 0 and the other extended to *m*<sup>1</sup> ≥ 1. Subtracted from (176), the part that is extended to *m*<sup>1</sup> ≥ 1, the remaining parts of (176) is just *C* in Eq. (163) (with *m*<sup>1</sup> ¼ 0, *m*<sup>2</sup> ≥1). Due to (175), we have

$$\begin{split} C &= \det(I + \Delta\_1 \Omega\_2) - \det(I + i\Delta\_1 \Delta\_2) \\ &= \sum\_{r=1}^{N} \sum\_{1 \le n\_1 < n\_2 < \cdots < n\_r \le N1 \le m\_1 < m\_2 < \cdots < m\_r \le N} \Omega\_1(n\_1, n\_2, \cdots, n\_r; 0, m\_2, \cdots, m\_r) \Omega\_2(0, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots, n\_r) \tag{177} \end{split} \tag{177}$$

$$\Omega\_1(n\_1, n\_2, \cdots, n\_r; 0, m\_2, \cdots m\_r) = \prod\_n \overline{h}\_n \prod\_m \mathcal{g}\_m \prod\_{n < n', m < m'} \left( \overline{\lambda\_n^2} - \overline{\lambda\_{n'}^2} \right) \left( \lambda\_{m'}^2 - \lambda\_m^2 \right) \prod\_{n, m} \frac{1}{\overline{\lambda\_n^2} - \lambda\_m^2} \tag{178}$$

$$
\Omega\_2(0, m\_2, \cdots, m\_r; n\_1, n\_2, \cdots n\_r) = \prod\_n \overline{g}\_n \prod\_m h\_m \prod\_{\substack{n < n', m < m' \\ \cdot \ \ \ \ \ \ \ \lambda\_m^2}} \left( \overline{\lambda\_n^2} - \overline{\lambda\_{n'}^2} \right) \left( \lambda\_{m'}^2 - \lambda\_m^2 \right) \prod\_{n, m} \frac{1}{\lambda\_m^2 - \overline{\lambda\_n^2}}, \tag{179}
$$

which leads to

$$\begin{split} \boldsymbol{\Omega}\_{1}(\boldsymbol{n}\_{1},\boldsymbol{n}\_{2},\cdots,\boldsymbol{n}\_{r};\boldsymbol{0},\boldsymbol{m}\_{2},\cdots\boldsymbol{m}\_{r})\boldsymbol{\Omega}\_{2}(\boldsymbol{0},\boldsymbol{m}\_{2},\cdots,\boldsymbol{m}\_{r};\boldsymbol{n}\_{1},\boldsymbol{n}\_{2},\cdots,\boldsymbol{n}\_{r}) \\ = (-1)^{r+1}\prod\_{n}e^{-\boldsymbol{\theta}\_{n}}e^{-i\boldsymbol{\theta}\_{n}}\prod\_{m}e^{-\boldsymbol{\theta}\_{m}}e^{i\boldsymbol{\rho}\_{m}}\prod\_{n$$

here *n*, *n*<sup>0</sup> ∈ð Þ *n*1, *n*2, ⋯, *nr* , *m*, *m*<sup>0</sup> ∈ ð Þ *m*2, ⋯, *mr* in (178)–(180). Finally, substituting (174) into (173), (180) into (177), and (173 and 177) into (162), we thus attain the explicit *N*-soliton solution to the DNLS equation with VBC under the reflectionless case, based on a pure Marchenko formalism and in no need of the concrete spectrum expression of *a*ð Þ*λ* . Obviously, the *N*-soliton solution permits uncertain complex constants *cn*<sup>0</sup> ð Þ *n* ¼ 1, 2, ⋯, *N* as well as an arbitrary global constant phase factor.

#### **3.5 The special examples for one- and two-soliton solutions**

In the case of one simple pole and one-soliton solution as *N* ¼ 1, according to (173), (177), (174), and (180), we have

$$\mathbf{C}\_{1} = \boldsymbol{\Omega}\_{1}(n\_{1} = 1; m\_{1} = 0)\boldsymbol{\Omega}\_{2}(m\_{1} = 0; n\_{1} = 1) = \overline{\mathbf{g}}\_{1}\overline{h}\_{1}(-1)^{1+1} = \overline{\mathbf{g}}\_{1}\overline{h}\_{1} \tag{181}$$

$$D\_1 = Q\_1(n\_1 = 1; m\_1 = 1)\\Q\_2(m\_1 = 1; m\_2 = 1) = 1 - \frac{\underline{g}\_1 \underline{h}\_1 \overline{\underline{h}}\_1 \underline{\lambda}\_1^2}{\left(\lambda\_1^2 - \overline{\lambda\_1^2}\right)^2} = 1 - \left|\underline{g}\_1 \underline{h}\_1\right|^2 \frac{\lambda\_1^2}{\left(\lambda\_1^2 - \overline{\lambda\_1^2}\right)^2} \tag{182}$$

From (167) and (168), we have (suppose *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>*1*e<sup>i</sup>β*<sup>1</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> <sup>þ</sup> *iv*<sup>1</sup> and *<sup>c</sup>*<sup>10</sup> <sup>¼</sup> *<sup>e</sup>*<sup>4</sup>*κ*1*x*<sup>10</sup> *<sup>e</sup><sup>i</sup>α*<sup>10</sup> )

$$\lg\_1 h\_1 = C\_1(t)e^{i2\dot{\lambda}\_1^2 \mathbf{x}} = c\_{10}e^{i4\dot{\lambda}\_1^4 t}e^{i2\dot{\lambda}\_1^2 \mathbf{x}} = e^{-\theta\_1}e^{i\rho\_1} \tag{183}$$

Up to a permitted constant global phase factor, the two-soliton solution gotten

above is actually equivalent to that gotten from both IST and Hirota's method [23, 24, 26, 27], verifying the validity of the algebraic techniques that is used and our formula of the generalized multi-soliton solution. Because Marchenko equations (128), (129), (144), and (145) had been strictly proved, the multi-soliton solution is certainly right as long as we correctly use the algebraic techniques, especially Binet-Cauchy formula for the principal minor expansion of some special matrices.

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

**4. Soliton solution of the DNLS equation based on Hirota's bilinear**

i*ut* þ *uxx* þ i j j *u*

i*ut* þ *uxx* þ i2j j *u*

which had been solved in Ref. [14] by using HBDT. We have paid special

where *f*, g are usually complex functions. Solution (191) is suitable for Eq. (190) and NLS equation, and so on, but not suitable for the DNLS equation. Just due to this fact, their work cannot deal with Eq. (189) at the same time. As is well known, rightly selecting an appropriate solution form is an important and key step to apply Hirota's bilinear derivative transform to an integrable equation like Eq. (189). Refs. [13, 16, 17, 23], etc., have proved the soliton solution of the DNLS equation must has

*u* ¼ *gf = f*

here and henceforth a bar over a letter represents complex conjugation. In view of the existing experiences of dealing with the DNLS equation, in the present section, we attempt to use the solution form (192) and HBDT to solve the DNLS equation. We demonstrate our solving approach step by step, and naturally

**4.1 Fundamental concepts and general properties of bilinear derivative**

For two differentiable functions *A x*ð Þ , *t* , *B x*ð Þ , *t* of two variables *x* and *t*, Hirota's

Bilinear derivative operator D had been found and defined in the early 1970s by Hirota R., a Japanese mathematical scientist [30–33]. Hirota's bilinear-derivative transform (HBDT for brevity) can be used to deal with some partial differential equation and to find some special solutions, such as soliton solutions and rogue wave solutions [26, 27, 32]. In this section, we use HBDT to solve DNLS equation with VBC and search for its soliton solution. The DNLS equation with VBC, that is,

> 2 *u*

> > 2

is one of the typical integrable nonlinear models, which is of a different form

*<sup>x</sup>* <sup>¼</sup> 0, (189)

*ux* ¼ 0, (190)

<sup>2</sup> (192)

*u* ¼ *g=f*, (191)

**derivative transform**

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

from the following equation:

following standard form

**transform**

**49**

attention to the following solution form in it [14]:

extend our conclusion to the *n*-soliton case in the end.

bilinear derivative operator, *D*, is defined as

$$\begin{aligned} \theta\_1 &= 4\mu\_1 v\_1 \left[ \mathbf{x} - \mathbf{x}\_{10} + 4 \left( \mu\_1^2 - v\_1^2 \right) t \right] \\ \varphi\_1 &= 2 \left( \mu\_1^2 - v\_1^2 \right) \mathbf{x} + \left[ 4 \left( \mu\_1^2 - v\_1^2 \right)^2 - 16 \mu\_1^2 v\_1^2 \right] \cdot t + a\_{10} \end{aligned} \tag{184}$$

Then from (181) and (182), we attain the one-soliton solution

$$u\_1(\mathbf{x}, t) = -2\frac{\mathbf{C}\_1 \overline{\mathbf{D}}\_1}{D\_1^2} = -2\left(1 - \frac{\lambda\_1^2}{\left(\lambda\_1^2 - \overline{\lambda\_1^2}\right)^2} e^{-2\theta\_1}\right) e^{-\theta\_1} e^{-i\rho\_1} \left/\left(1 - \frac{\overline{\lambda\_1^2}}{\left(\lambda\_1^2 - \overline{\lambda\_1^2}\right)^2} e^{-2\theta\_1}\right)^2 \tag{185}$$

By further redefinition of its soliton center and initial phase, the single soliton solution can be further rewritten as usual standard form. It is easy to find, up to a permitted well-known constant global phase factor, the one-soliton solution to DNLS equation gotten in the pure Marchenko formalism is in perfectly agreement with that gotten from other approaches [23, 24, 26, 27].

As *N* ¼ 2 in the case of two-soliton solution corresponding to double simple poles, we have

*<sup>u</sup>*2ð Þ¼� *<sup>x</sup>*, *<sup>t</sup>* <sup>2</sup>*C*2*D*2*=D*<sup>2</sup> <sup>2</sup> (186) *<sup>C</sup>*<sup>2</sup> <sup>¼</sup> <sup>X</sup> *n*1¼1, 2 *m*1¼0 Ω1ð Þ *n*1, 0 Ω2ð Þþ 0; *n*<sup>1</sup> X *n*1¼1, *n*2¼2 *m*1¼0, *m*2¼1, 2 Ω1ð Þ *n*1, *n*2; 0, *m*<sup>2</sup> Ω2ð Þ 0, *m*2; n1, *n*<sup>2</sup> ¼ Ω1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 0 Ω2ð*m*<sup>1</sup> ¼ 0; *n*<sup>1</sup> ¼ 1Þ þ Ω1ð Þ *n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 0 Ω2ð Þ *m*<sup>1</sup> ¼ 0; *n*<sup>1</sup> ¼ 2 þΩ1ð Þ *n*<sup>1</sup> ¼ 1, *n*<sup>2</sup> ¼ 2; *m*<sup>1</sup> ¼ 0, *m*<sup>2</sup> ¼ 1 Ω2ð Þ *m*<sup>1</sup> ¼ 0, *m*<sup>2</sup> ¼ 1; *n*<sup>2</sup> ¼ 1, *n*<sup>2</sup> ¼ 2 þΩ1ð Þ *n*<sup>1</sup> ¼ 1, *n*<sup>2</sup> ¼ 2; *m*<sup>1</sup> ¼ 0, *m*<sup>2</sup> ¼ 2 Ω2ð Þ *m*<sup>1</sup> ¼ 0, *m*<sup>2</sup> ¼ 2; *n*<sup>1</sup> ¼ 1, *n*<sup>2</sup> ¼ 2 ¼ *g*1*h*<sup>1</sup> þ *g*2*h*<sup>2</sup> � *g*1*h*<sup>1</sup> � � � � 2 *g*2*h*<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *λ*2 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *<sup>g</sup>*2*h*<sup>2</sup> � � � � 2 *g*1*h*<sup>1</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *λ*2 2 *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> ¼ *e* �*θ e* �*iφ*<sup>1</sup> <sup>þ</sup> *<sup>e</sup>* �*θ*<sup>2</sup> *e* �*iφ*<sup>2</sup> � *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *λ*2 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *<sup>e</sup>* �2*θ*1�*θ*<sup>2</sup> *e* �*iφ*<sup>2</sup> � *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *λ*2 2 *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *<sup>e</sup>* �2*θ*2�*θ*<sup>1</sup> *e* �*iφ*<sup>1</sup> (187)

*<sup>D</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>X<sup>2</sup> *r*¼1 X 1 ≤*n*<sup>1</sup> <*n*<sup>2</sup> ≤2 X 1 ≤ *m*<sup>1</sup> < *m*<sup>2</sup> ≤2 *Q*<sup>1</sup> *n*1, *n*2, ⋯, *nr* ð Þ ; *m*1, *m*2, ⋯, *mr Q*<sup>2</sup> *m*1, *m*2, ⋯, *mr* ð Þ ; *n*1, *n*2, ⋯, *nr* ¼ 1 þ *Q*1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 1 *Q*2ð*m*<sup>1</sup> ¼ 1, *n*<sup>1</sup> ¼ 1Þ þ *Q*1ð Þ *n*<sup>1</sup> ¼ 1; *m*<sup>1</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 2; *n*<sup>1</sup> ¼ 1 þ*Q*1ð Þ *n*<sup>1</sup> ¼ 2; *m*<sup>1</sup> ¼ 1 *Q*2ð*m*<sup>1</sup> ¼ 1, *n*<sup>1</sup> ¼ 2Þ þ *Q*1ð Þ *n*<sup>1</sup> ¼ 2, *m*<sup>1</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 2; *n*<sup>1</sup> ¼ 2 þ*Q*1ð Þ *n*<sup>1</sup> ¼ 1, *n*<sup>2</sup> ¼ 2; *m*<sup>1</sup> ¼ 1, *m*<sup>2</sup> ¼ 2 *Q*2ð Þ *m*<sup>1</sup> ¼ 1, *m*<sup>2</sup> ¼ 2; *n*<sup>1</sup> ¼ 1, *n*<sup>2</sup> ¼ 2 ¼ 1 � *g*1*h*<sup>1</sup> � � � � <sup>2</sup> *λ*<sup>2</sup> 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> � *<sup>g</sup>*2*h*<sup>2</sup> � � � � <sup>2</sup> *λ*<sup>2</sup> 2 *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> � *<sup>g</sup>*1*h*1*g*2*h*<sup>2</sup> *λ*2 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> �*g*2*h*2*g*1*h*<sup>1</sup> *λ*2 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> <sup>þ</sup> *<sup>g</sup>*1*h*<sup>1</sup> � � � �<sup>2</sup> *<sup>g</sup>*2*h*<sup>2</sup> � � � �<sup>2</sup> *<sup>λ</sup>*<sup>2</sup> 1*λ*2 <sup>2</sup> *λ*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *<sup>e</sup>* �2*θ*<sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *<sup>e</sup>* �2*θ*<sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *<sup>e</sup>* �*θ*1�*θ*<sup>2</sup> *e i φ*2�*φ*<sup>1</sup> ð Þ � *<sup>λ</sup>*<sup>2</sup> 2 *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *<sup>e</sup>* �*θ*1�*θ*<sup>2</sup> *e <sup>i</sup> <sup>φ</sup>*1�*φ*<sup>2</sup> ð Þ <sup>þ</sup> *λ*2 1*λ*2 <sup>2</sup> *λ*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*<sup>2</sup> <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> *λ*2 <sup>2</sup> � *<sup>λ</sup>*<sup>2</sup> 2 � �<sup>2</sup> *<sup>e</sup>* �2ð Þ *θ*1þ*θ*<sup>2</sup> (188)

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

Up to a permitted constant global phase factor, the two-soliton solution gotten above is actually equivalent to that gotten from both IST and Hirota's method [23, 24, 26, 27], verifying the validity of the algebraic techniques that is used and our formula of the generalized multi-soliton solution. Because Marchenko equations (128), (129), (144), and (145) had been strictly proved, the multi-soliton solution is certainly right as long as we correctly use the algebraic techniques, especially Binet-Cauchy formula for the principal minor expansion of some special matrices.

## **4. Soliton solution of the DNLS equation based on Hirota's bilinear derivative transform**

Bilinear derivative operator D had been found and defined in the early 1970s by Hirota R., a Japanese mathematical scientist [30–33]. Hirota's bilinear-derivative transform (HBDT for brevity) can be used to deal with some partial differential equation and to find some special solutions, such as soliton solutions and rogue wave solutions [26, 27, 32]. In this section, we use HBDT to solve DNLS equation with VBC and search for its soliton solution. The DNLS equation with VBC, that is,

$$\left(\mathbf{u}\_t + \mathbf{u}\_{\infty} + \mathbf{i}\left(|u|^2 u\right)\_{\mathbf{x}} = \mathbf{0},\tag{189}$$

is one of the typical integrable nonlinear models, which is of a different form from the following equation:

$$\left|\mathbf{u}\_t + \mathbf{u}\_{\infty} + \mathbf{i}\mathbf{2}|u|^2 u\_{\infty} = \mathbf{0},\tag{190}$$

which had been solved in Ref. [14] by using HBDT. We have paid special attention to the following solution form in it [14]:

$$
\mathfrak{u} = \mathfrak{g}/\mathfrak{f},\tag{191}
$$

where *f*, g are usually complex functions. Solution (191) is suitable for Eq. (190) and NLS equation, and so on, but not suitable for the DNLS equation. Just due to this fact, their work cannot deal with Eq. (189) at the same time. As is well known, rightly selecting an appropriate solution form is an important and key step to apply Hirota's bilinear derivative transform to an integrable equation like Eq. (189). Refs. [13, 16, 17, 23], etc., have proved the soliton solution of the DNLS equation must has following standard form

$$
\mu = \overline{\text{gf}} / f^2 \tag{192}
$$

here and henceforth a bar over a letter represents complex conjugation.

In view of the existing experiences of dealing with the DNLS equation, in the present section, we attempt to use the solution form (192) and HBDT to solve the DNLS equation. We demonstrate our solving approach step by step, and naturally extend our conclusion to the *n*-soliton case in the end.

### **4.1 Fundamental concepts and general properties of bilinear derivative transform**

For two differentiable functions *A x*ð Þ , *t* , *B x*ð Þ , *t* of two variables *x* and *t*, Hirota's bilinear derivative operator, *D*, is defined as

*Nonlinear Optics - From Solitons to Similaritons*

$$D\_t^n D\_{\mathbf{x}'}^m A \cdot B = \left(\frac{\partial}{\partial t} - \frac{\partial}{\partial t'}\right)^n \left(\frac{\partial}{\partial \mathbf{x}} - \frac{\partial}{\partial \mathbf{x}'}\right)^m A\left(\mathbf{x}, t\right) B(\mathbf{x}', t')|\_{t'=t, \mathbf{x}'=\mathbf{x}}\tag{193}$$

which is different from the usual derivative, for example,

$$\begin{aligned} D\_{\mathbf{x}}A \cdot \mathbf{B} &= A\_{\mathbf{x}}B - AB\_{\mathbf{x}} \\ D\_{\mathbf{x}}^2 A \cdot \mathbf{B} &= A\_{\mathbf{x}\mathbf{x}}B - 2A\_{\mathbf{x}}B\_{\mathbf{x}} + AB\_{\mathbf{x}\mathbf{x}} \\ D\_{\mathbf{x}}^3 A \cdot \mathbf{B} &= A\_{\mathbf{x}\mathbf{x}}B - 3A\_{\mathbf{x}\mathbf{x}}B\_{\mathbf{x}} + 3A\_{\mathbf{x}}B\_{\mathbf{x}\mathbf{x}} - AB\_{\mathbf{x}\mathbf{x}} \end{aligned} \tag{194}$$

*<sup>f</sup> <sup>f</sup>* <sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup>

follows:

parameter *ε*

*x*

� �*<sup>g</sup>* � *<sup>f</sup>* � *gf* <sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

*x* � �*<sup>f</sup>* � *<sup>f</sup>* <sup>þ</sup> *<sup>f</sup>*

�2 *Dx f*

We can extract the needed bilinear derivative equations from Eq. (203) as

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x*

Functions *g x*ð Þ , *t* , *f x*ð Þ , *t* can be expanded, respectively, as series of a small

*<sup>g</sup>* <sup>¼</sup> <sup>X</sup> *i εi*

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup><sup>X</sup>

terms with the same orders of *ε* at two sides of (204)–(206), we attain

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *f*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *<sup>g</sup>*ð Þ<sup>2</sup> � <sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>1</sup> � *<sup>f</sup>*

ð Þ<sup>2</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

ð Þ<sup>3</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

ð Þ 4 þ *f* ð Þ<sup>3</sup> � *<sup>f</sup>* ð Þ1 þ *f* ð Þ<sup>1</sup> � *<sup>f</sup>* ð Þ3 þ *f* ð Þ<sup>2</sup> � *<sup>f</sup>*

ð Þ 4 þ *f* ð Þ<sup>3</sup> � *<sup>f</sup>* ð Þ1 þ *f* ð Þ<sup>1</sup> � *<sup>f</sup>* ð Þ3 þ *f* ð Þ<sup>2</sup> � *<sup>f</sup>*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *f*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *<sup>g</sup>*ð Þ<sup>3</sup> � <sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>2</sup> � *<sup>f</sup>*

ð Þ3 þ *f* ð Þ<sup>2</sup> � *<sup>f</sup>* ð Þ1 þ *f* ð Þ<sup>1</sup> � *<sup>f</sup>* ð Þ<sup>2</sup> � � <sup>¼</sup> <sup>i</sup> *<sup>g</sup>*ð Þ<sup>2</sup> *<sup>g</sup>*ð Þ<sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>1</sup> *<sup>g</sup>*ð Þ<sup>2</sup> � �*=*2 (217)

ð Þ <sup>4</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

ð Þ <sup>4</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

*Dx f*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *f*

ð Þ<sup>3</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *<sup>g</sup>*ð Þ <sup>4</sup> � <sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>3</sup> � *<sup>f</sup>*

*Dx f*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *<sup>g</sup>*ð Þ<sup>5</sup> � <sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ <sup>4</sup> � *<sup>f</sup>*

*Dx f*

<sup>i</sup>*Dt* <sup>þ</sup> *<sup>D</sup>*<sup>2</sup> *x* � � *f*

**51**

*Dx f*

<sup>i</sup>*∂<sup>t</sup>* <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup> *x* *i εi f*

ð Þ<sup>1</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

ð Þ2 þ *f* ð Þ<sup>1</sup> � *<sup>f</sup>* ð Þ<sup>1</sup> � � <sup>¼</sup> <sup>0</sup> (213)

ð Þ<sup>1</sup> � � <sup>¼</sup> <sup>i</sup>*g*ð Þ<sup>1</sup> � *<sup>g</sup>*ð Þ<sup>1</sup> *<sup>=</sup>*2 (214)

ð Þ<sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>1</sup> � *<sup>f</sup>* ð Þ<sup>2</sup> � � <sup>¼</sup> 0 (215)

ð Þ<sup>2</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>1</sup> � *<sup>f</sup>* ð Þ<sup>3</sup> � � <sup>¼</sup> 0 (218)

ð Þ<sup>2</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>2</sup> � *<sup>f</sup>* ð Þ<sup>3</sup> � � <sup>¼</sup> 0 (221)

ð Þ<sup>2</sup> � � <sup>¼</sup> 0 (219)

<sup>¼</sup> <sup>i</sup> *<sup>g</sup>*ð Þ<sup>3</sup> *<sup>g</sup>*ð Þ<sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>2</sup> *<sup>g</sup>*ð Þ<sup>2</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>1</sup> *<sup>g</sup>*ð Þ<sup>3</sup> � �*=*<sup>2</sup> (220)

Substituting (207) and (208) into (204)–(206) and equating the sum of the

ð Þ<sup>1</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

ð Þ<sup>2</sup> � <sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>f</sup>*

ð Þ2 þ *f* ð Þ<sup>1</sup> � *<sup>f</sup>*

ð Þ3 þ *f* ð Þ<sup>2</sup> � *<sup>f</sup>* ð Þ1 þ *f* ð Þ<sup>1</sup> � *<sup>f</sup>* ð Þ<sup>2</sup> � � <sup>¼</sup> 0 (216)

ð Þ<sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>2</sup> � *<sup>f</sup>*

ð Þ<sup>1</sup> <sup>þ</sup> *<sup>g</sup>*ð Þ<sup>3</sup> � *<sup>f</sup>*

ð Þ<sup>2</sup> � �

<sup>3</sup> � *<sup>g</sup>* <sup>2</sup>*Dxf* � *<sup>f</sup>* � <sup>i</sup>*gg*

� �*<sup>g</sup>* � *<sup>f</sup>* <sup>¼</sup> <sup>0</sup> (204)

� �*<sup>f</sup>* � *<sup>f</sup>* <sup>¼</sup> <sup>0</sup> (205)

*Dxf* � *f* ¼ i*gg=*2 (206)

� �*g*ð Þ<sup>1</sup> <sup>¼</sup> <sup>0</sup> (209)

ð Þ<sup>1</sup> � � <sup>¼</sup> <sup>0</sup> (211)

ð Þ<sup>1</sup> � � <sup>¼</sup> <sup>0</sup> (210)

ð Þ<sup>1</sup> � � <sup>¼</sup> <sup>0</sup> (212)

*g*ð Þ*<sup>i</sup>* (207)

ð Þ*<sup>i</sup>* (208)

h i � � <sup>¼</sup> 0 (203)

where *A x*ð Þ , *t* , *B x*ð Þ , *t* are two functions derivable for an arbitrary order, and the dot � between them represents a kind of ordered product. Hirota's bilinear derivative has many interesting properties. Some important properties to be used afterwards are listed as follows:

$$\left(\left\|D\_t^n D\_x^m A \cdot B\right\| = (-1)^{n+m} D\_t^n D\_x^m B \cdot A\right) \tag{195}$$

for example, *DxA* � *<sup>B</sup>* ¼ �*DxB* � *<sup>A</sup>*; *DxA* � *<sup>A</sup>* <sup>¼</sup> 0; *<sup>D</sup>*<sup>2</sup> *xA* � *<sup>B</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> *xB* � *<sup>A</sup>*; *<sup>D</sup><sup>n</sup> xA* � 1 ¼ *∂n xA*; *D<sup>n</sup> <sup>x</sup>*<sup>1</sup> � *<sup>A</sup>* ¼ �ð Þ<sup>1</sup> *<sup>n</sup> ∂n xA*

$$\otimes D\_{\mathbf{x}}^{n}\mathbf{A} \cdot \mathbf{B} = D\_{\mathbf{x}}^{n-m}D\_{\mathbf{x}}^{m}\mathbf{A} \cdot \mathbf{B}, (m < n) \tag{196}$$

➂ Suppose *η<sup>i</sup>* ¼ Ω*<sup>i</sup> t* þ Λ*ix* þ *η*0*<sup>i</sup>*, *i* ¼ 1, 2, *Ωi*, Λ*i*, *η*0*<sup>i</sup>* are complex constants, then

$$D\_t^n D\_x^m \exp\left(\eta\_1\right) \cdot \exp\left(\eta\_2\right) = \left(\mathcal{Q}\_1 - \mathcal{Q}\_2\right)^n \left(\Lambda\_1 - \Lambda\_2\right)^m \exp\left(\eta\_1 + \eta\_2\right) \tag{197}$$

Especially, we have *Dn <sup>t</sup> Dm <sup>x</sup>* exp *η*<sup>1</sup> ð Þ� exp *η*<sup>2</sup> ð Þ¼ 0 as *Ω*<sup>1</sup> ¼ *Ω*<sup>2</sup> or Λ<sup>1</sup> ¼ Λ2. Some other important properties are listed in the Appendix.

#### **4.2 Bilinear derivative transform of DNLS equation**

After a suitable solution form, for example, (192) has been selected, under the Hirota's bilinear derivative transform, a partial differential equation like (189) can be generally changed into [20, 26, 27].

$$F\_1(D\_t, D\_x \cdots) \mathfrak{g}\_1 \cdot f\_1 + F\_2(D\_t, D\_x \cdots) \mathfrak{g}\_2 \cdot f\_2 = \mathbf{0} \tag{198}$$

where *Fi Dt* ð Þ , *Dx*⋯ , *i* ¼ 1, 2 are the polynomial functions of *Dt*, *Dx* ⋯; and *gi* , *fi* , *i* ¼ 1, 2, are the differentiable functions of two variables *x* and *t* . Using formulae in the Appendix and properties ①–③ of bilinear derivative transform numerated in the last chapter, with respect to (192), we have

$$\mu\_t = \mathrm{D}\_t \overline{\mathbf{g}}^\prime \cdot f^2 / f^4 = \left[ f \, \overline{f} \mathrm{D}\_t \mathbf{g} \cdot f - \mathbf{g} \mathrm{D}\_t \mathbf{f} \cdot \overline{\mathbf{f}} \right] / f^4 \tag{199}$$

$$\mu\_{\mathbf{x}} = \mathbf{D}\_{\mathbf{x}} \mathbf{g} \overline{\mathbf{f}} \cdot f^2 / f^4 = \left[ f \, \overline{f} \mathbf{D}\_{\mathbf{x}} \mathbf{g} \cdot f - \mathbf{g} \mathbf{f} \mathbf{D}\_{\mathbf{x}} \mathbf{f} \cdot \overline{f} \right] / f^4 \tag{200}$$

$$\mu\_{\text{xx}} = \left[ f \, \overline{f} D\_{\text{x}}^2 \mathbf{g} \cdot f - \mathbf{2} (D\_{\text{x}} \mathbf{g} \cdot f) \left( D\_{\text{x}} f \cdot \overline{f} \right) + \mathbf{g} f \mathbf{D}\_{\text{x}}^2 f \cdot \overline{f} \right] / f^4 - 2 \overline{\mathbf{g}} \, \mathbf{D}\_{\text{x}}^2 f \cdot f / f^4 \tag{201}$$

$$\left(\left|u\right|^{2}u\right)\_{\mathbf{x}} = \left[2\mathbf{g}\overline{\mathbf{g}}D\_{\mathbf{x}}\mathbf{g}\cdot f - \mathbf{g}^{2}(\overline{\mathbf{g}}f)\_{\mathbf{x}}\right]/f^{4} \tag{202}$$

Substituting the above expressions (199)–(202) into Eq. (189), the latter can be reduced to [26, 27].

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

$$\frac{1}{2}f\overline{f}\left(\mathrm{i}D\_{\mathrm{t}}+D\_{\mathrm{x}}^{2}\right)\mathbf{g}\cdot f - \mathrm{g}f\left(\mathrm{i}D\_{\mathrm{t}}+D\_{\mathrm{x}}^{2}\right)f\cdot\overline{f} + f^{-2}D\_{\mathrm{x}}f^{3}\cdot\left[\mathrm{g}\left(2D\_{\mathrm{x}}f\cdot\overline{f}-\mathrm{i}\overline{\mathrm{g}}\right)\right] = \mathbf{0} \tag{203}$$

We can extract the needed bilinear derivative equations from Eq. (203) as follows:

$$(\mathbf{i}D\_t + D\_x^2)\mathbf{g} \cdot f = \mathbf{0} \tag{204}$$

$$(\mathrm{i}D\_t + D\_x^2)\overline{f} \cdot \overline{f} = \mathbf{0} \tag{205}$$

$$D\_{\mathbf{x}}f \cdot \overline{f} = \mathbf{i} \mathbf{g} \overline{\mathbf{g}}/2 \tag{206}$$

Functions *g x*ð Þ , *t* , *f x*ð Þ , *t* can be expanded, respectively, as series of a small parameter *ε*

$$\mathbf{g} = \sum\_{i} \mathbf{e}^{i} \mathbf{g}^{(i)} \tag{207}$$

$$f = \mathbf{1} + \sum\_{i} \varepsilon^{i} f^{(i)} \tag{208}$$

Substituting (207) and (208) into (204)–(206) and equating the sum of the terms with the same orders of *ε* at two sides of (204)–(206), we attain

$$(\mathbf{i}\partial\_t + \partial\_x^2)\mathbf{g}^{(1)} = \mathbf{0} \tag{209}$$

$$\left(\mathrm{i}D\_t + D\_x^2\right)\left(f^{(1)} \cdot \mathbf{1} + \mathbf{1} \cdot \overline{f}^{(1)}\right) = \mathbf{0} \tag{210}$$

$$D\_{\mathbf{x}}\left(f^{(1)}\cdot\mathbf{1} + \mathbf{1}\cdot\overline{f}^{(1)}\right) = \mathbf{0} \tag{211}$$

$$\left(\mathbf{i}D\_t + D\_x^2\right)\left(\mathbf{g}^{(2)} \cdot \mathbf{1} + \mathbf{g}^{(1)} \cdot f^{(1)}\right) = \mathbf{0} \tag{212}$$

$$\left(\left(\mathrm{i}D\_t + D\_x^2\right)\left(f^{(2)} \cdot \mathbf{1} + \mathbf{1} \cdot \overline{f}^{(2)} + f^{(1)} \cdot \overline{f}^{(1)}\right) = \mathbf{0} \tag{213}$$

$$D\_{\mathbf{x}}\left(\boldsymbol{f}^{(2)}\cdot\mathbf{1}+\mathbf{1}\cdot\overline{\boldsymbol{f}}^{(2)}+\boldsymbol{f}^{(1)}\cdot\overline{\boldsymbol{f}}^{(1)}\right)=\mathbf{i}\mathbf{g}^{(1)}\cdot\overline{\mathbf{g}}^{(1)}/2\tag{214}$$

$$\left(\mathrm{i}D\_t + D\_x^2\right)\left(\mathrm{g}^{(3)} \cdot \mathbf{1} + \mathrm{g}^{(2)} \cdot f^{(1)} + \mathrm{g}^{(1)} \cdot f^{(2)}\right) = \mathbf{0} \tag{215}$$

$$\left(\mathrm{i}D\_{t} + D\_{\mathrm{x}}^{2}\right)\left(f^{(3)} \cdot \mathbf{1} + \mathbf{1} \cdot \overline{f}^{(3)} + f^{(2)} \cdot \overline{f}^{(1)} + f^{(1)} \cdot \overline{f}^{(2)}\right) = \mathbf{0} \tag{216}$$

$$D\_x \left( f^{(3)} \cdot \mathbf{1} + \mathbf{1} \cdot \overline{f}^{(3)} + f^{(2)} \cdot \overline{f}^{(1)} + f^{(1)} \cdot \overline{f}^{(2)} \right) = \mathbf{i} \left( \mathbf{g}^{(2)} \overline{\mathbf{g}}^{(1)} + \mathbf{g}^{(1)} \overline{\mathbf{g}}^{(2)} \right) / 2 \tag{217}$$

$$\left(\mathrm{i}D\_{\mathrm{f}} + D\_{\mathrm{x}}^{2}\right)\left(\mathrm{g}^{(4)} \cdot \mathbf{1} + \mathrm{g}^{(3)} \cdot f^{(1)} + \mathrm{g}^{(2)} \cdot f^{(2)} + \mathrm{g}^{(1)} \cdot f^{(3)}\right) = \mathbf{0} \tag{218}$$

$$\left(\mathbf{i}\mathcal{D}\_t + \mathcal{D}\_x^2\right)\left(f^{(4)}\cdot\mathbf{1} + \mathbf{1}\cdot\overline{f}^{(4)} + f^{(3)}\cdot\overline{f}^{(1)} + f^{(1)}\cdot\overline{f}^{(3)} + f^{(2)}\cdot\overline{f}^{(2)}\right) = \mathbf{0} \tag{219}$$

$$\begin{aligned} D\_x \left( f^{(4)} \cdot \mathbf{1} + \mathbf{1} \cdot \overline{f}^{(4)} + f^{(3)} \cdot \overline{f}^{(1)} + f^{(1)} \cdot \overline{f}^{(3)} + f^{(2)} \cdot \overline{f}^{(2)} \right) \\ = \mathbf{i} \left( \mathbf{g}^{(3)} \overline{\mathbf{g}}^{(1)} + \mathbf{g}^{(2)} \overline{\mathbf{g}}^{(2)} + \mathbf{g}^{(1)} \overline{\mathbf{g}}^{(3)} \right) / 2 \end{aligned} \tag{220}$$

$$\left(\mathrm{i}D\_t + D\_x^2\right)\left(\mathrm{g}^{(5)} \cdot \mathbf{1} + \mathrm{g}^{(4)} \cdot f^{(1)} + \mathrm{g}^{(3)} \cdot f^{(2)} + \mathrm{g}^{(2)} \cdot f^{(3)}\right) = \mathbf{0} \tag{221}$$

$$\left(\left(\mathrm{i}D\_t + D\_x^2\right)\left(f^{(5)} \cdot \mathbf{1} + \mathbf{1} \cdot \overline{f}^{(5)} + f^{(4)} \cdot \overline{f}^{(1)} + f^{(1)} \cdot \overline{f}^{(4)} + f^{(3)} \cdot \overline{f}^{(2)} + f^{(2)} \cdot \overline{f}^{(3)}\right) = \mathbf{0} \tag{222}$$

$$\begin{aligned} D\_x \left( f^{(5)} \cdot \mathbf{1} + \mathbf{1} \cdot \overline{f}^{(5)} + f^{(4)} \cdot \overline{f}^{(1)} + f^{(1)} \cdot \overline{f}^{(4)} + f^{(3)} \cdot \overline{f}^{(2)} + f^{(2)} \cdot \overline{f}^{(3)} \right) \\ = \mathbf{i} \left( \mathbf{g}^{(4)} \overline{\mathbf{g}}^{(1)} + \mathbf{g}^{(3)} \overline{\mathbf{g}}^{(2)} + \mathbf{g}^{(2)} \overline{\mathbf{g}}^{(3)} + \mathbf{g}^{(1)} \overline{\mathbf{g}}^{(4)} \right) / 2 \end{aligned} \tag{223}$$

The above equations, (209)–(223), contain the whole information needed to search for a soliton solution of the DNLS equation with VBC.

#### **4.3 Soliton solution of the DNLS equation with VBC based on HBDT**

#### *4.3.1 One-soliton solution*

For the one-soliton case, due to (209)–(211) and considering the transform property ③, we can select *g*ð Þ<sup>1</sup> and *f* ð Þ<sup>1</sup> respectively as

$$\mathbf{g}^{(1)} = \mathbf{e}^{\eta\_1}, \eta\_1 = \mathcal{Q}\_1 \mathbf{t} + \Lambda\_1 \mathbf{x} + \eta\_{10}, \mathcal{Q}\_1 = \mathbf{i}\Lambda\_1^2,\tag{224}$$

$$f^{(1)} = \mathbf{0} \tag{225}$$

series (207) and (208) have been successfully cut off to have limited terms as

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

i 2

*f* <sup>1</sup> ¼ 1 þ

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

we attain the one-soliton solution to the DNLS equation with VBC

<sup>1</sup> *<sup>t</sup>*þ*η*<sup>10</sup> *<sup>=</sup>*<sup>2</sup> � <sup>e</sup>�<sup>Θ</sup>1e<sup>i</sup>Φ<sup>1</sup> ; e*<sup>η</sup>*<sup>10</sup> *<sup>=</sup>*<sup>2</sup> � <sup>e</sup><sup>4</sup>*μ*1*ν*1*x*<sup>10</sup> ei*<sup>α</sup>*<sup>10</sup> ; <sup>Θ</sup><sup>1</sup> � <sup>4</sup>*μ*1*v*<sup>1</sup> *<sup>x</sup>* � *<sup>x</sup>*<sup>10</sup> <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

> 2 *x*�i4*λ* 4

Λ1 Λ<sup>1</sup> þ Λ<sup>1</sup> � �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*<sup>1</sup>

1

CA

It is easy to find, up to a permitted constant global phase factor e<sup>i</sup><sup>π</sup> ¼ �1, the one-soliton solution (234) or (237) gotten in this paper is in perfect agreement with that gotten from other approaches [16]. By further redefining its soliton center, initial phase and *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>*1e<sup>i</sup>*β*<sup>1</sup> , the one-soliton solution can be changed into the usual

On the other hand, just like in Ref. [13], we can rewrite *g*<sup>1</sup> and *f* <sup>1</sup> in a more

e�<sup>Θ</sup>1e�iΦ<sup>1</sup>

� �e�<sup>Θ</sup>1e�iΦ1*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>e</sup>�i2*<sup>β</sup>*1e�2Θ<sup>1</sup>

,

0

B@

*<sup>g</sup>*<sup>1</sup> <sup>¼</sup> <sup>e</sup>*<sup>η</sup>*1þ*φ*<sup>1</sup> (239)

*<sup>f</sup>* <sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup> *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ0 *<sup>θ</sup>*11<sup>0</sup> (240)

<sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> <sup>e</sup>�2Θ<sup>1</sup>

� �<sup>2</sup> (238)

**Figure 1**. If we redefine the parameter Λ<sup>1</sup> as Λ<sup>1</sup> � �i2*λ*

� <sup>2</sup> *<sup>μ</sup>*<sup>2</sup>

<sup>1</sup> � *<sup>v</sup>*<sup>2</sup> 1 � �*<sup>x</sup>* <sup>þ</sup> <sup>4</sup> *<sup>μ</sup>*<sup>2</sup>

*<sup>g</sup>*<sup>1</sup> <sup>¼</sup> <sup>e</sup>*<sup>η</sup>*<sup>1</sup> � <sup>e</sup>�i2*<sup>λ</sup>*

i 2

2

*<sup>λ</sup>*<sup>2</sup> � *<sup>λ</sup>* <sup>2</sup> � �<sup>2</sup> <sup>e</sup>�2<sup>Θ</sup>

1

*f* <sup>1</sup> ¼ 1 þ

<sup>¼</sup> <sup>1</sup> � *<sup>λ</sup>*

*λ*2 <sup>1</sup> � *<sup>λ</sup>*<sup>2</sup> 1 � �<sup>2</sup> <sup>e</sup>�2Θ<sup>1</sup>

<sup>¼</sup> 2 1 � *<sup>λ</sup>*<sup>2</sup>

*<sup>u</sup>*1ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>4</sup>∣*λ*1∣sin 2*β*<sup>1</sup> <sup>1</sup> <sup>þ</sup> ei2*<sup>β</sup>*1e�2Θ<sup>1</sup>

0

B@

*g*<sup>1</sup> ¼ *e*

Λ1 Λ<sup>1</sup> þ Λ<sup>1</sup>

where *ε<sup>i</sup>* has been absorbed into the constant *e<sup>η</sup>*<sup>10</sup> by redefiniing *η*10. In the end,

2

<sup>1</sup> � *<sup>v</sup>*<sup>2</sup> 1

*<sup>t</sup>*þ*η*<sup>10</sup> <sup>¼</sup> 2e�<sup>Θ</sup>1e�iΦ<sup>1</sup>

<sup>1</sup> � *<sup>v</sup>*<sup>2</sup> 1 � �<sup>2</sup> � <sup>16</sup>*μ*<sup>2</sup>

2

1*v*2 1 h i *<sup>t</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>10</sup> (235)

*u*1ð Þ¼ *x*, *t g*1*f* <sup>1</sup>*= f*

which is characterized with two complex parameters Λ<sup>1</sup> and *η*<sup>10</sup> and shown in

� �*t* � �; Φ<sup>1</sup>

*<sup>η</sup>*<sup>1</sup> (232)

<sup>1</sup> (234)

<sup>1</sup> and *λ*<sup>1</sup> � *μ*<sup>1</sup> þ i*v*1, then

(236)

1

2

CA

(237)

� �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*<sup>1</sup> (233)

follows:

ei2*<sup>λ</sup>*<sup>2</sup> 1*x*þi4*λ*<sup>4</sup>

*<sup>u</sup>*1ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>g</sup>*1*<sup>f</sup>* <sup>1</sup>

*f* 2 1

typical form [16, 23, 26, 27].

appropriate or "standard" form

**53**

Then

From (212), one can select *<sup>g</sup>*ð Þ<sup>2</sup> <sup>¼</sup> 0. From (214), we can attain

$$f^{(2)} - \overline{f}^{(2)} = \frac{\mathbf{i}}{2} \frac{\mathbf{1}}{\Lambda\_1 + \overline{\Lambda\_1}} \mathbf{e}^{\eta\_1 + \overline{\eta}\_1} \tag{226}$$

where the vanishing boundary condition, *u* ! 0 as j j *x* ! ∞, is used. Then

$$\partial\_t \left( f^{(2)} - \overline{f}^{(2)} \right) = -\frac{\mathbf{1} \Lambda\_1^2 - \overline{\Lambda}\_1^2}{2 \Lambda\_1 + \overline{\Lambda}\_1} \mathbf{e}^{\eta\_1 + \overline{\eta}\_1} \tag{227}$$

Substituting (226) and (227) into Eq. (213), we can attain

$$f^{(2)} + \overline{f}^{(2)} = \frac{\mathbf{i}}{2} \frac{\Lambda\_1 - \overline{\Lambda}\_1}{\left(\Lambda\_1 + \overline{\Lambda}\_1\right)^2} \mathbf{e}^{\eta\_1 + \overline{\eta}\_1} \tag{228}$$

From (226) and (228), we can get an expression of *f* ð Þ2

$$f^{(2)} = \frac{\mathbf{i}}{2} \frac{\Lambda\_1}{\left(\Lambda\_1 + \overline{\Lambda}\_1\right)^2} \mathbf{e}^{\eta\_1 + \overline{\eta}\_1} \tag{229}$$

Due to (224) and (229), we can also easily verify that

$$(\mathbf{i}D\_t + D\_x^2)\mathbf{g}^{(1)} \cdot f^{(2)} = \mathbf{0} \tag{230}$$

which immediately leads to

$$(\mathbf{i}\partial\_t + \partial\_x^2)\mathbf{g}^{(3)} = \mathbf{0} \tag{231}$$

in Eq. (215). Then from (215), we can select *<sup>g</sup>*ð Þ<sup>3</sup> <sup>¼</sup> 0. For the same reason, from (216)–(223), we can select *f* ð Þ<sup>3</sup> , *g*ð Þ <sup>4</sup> , *g*ð Þ<sup>5</sup> , … ; *f* ð Þ <sup>4</sup> , *f* ð Þ<sup>5</sup> , … all to be zero. Thus the

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

series (207) and (208) have been successfully cut off to have limited terms as follows:

$$\mathbf{g}\_1 = e^{\eta\_1} \tag{232}$$

$$f\_1 = \mathbf{1} + \frac{\mathbf{i}}{2} \frac{\Lambda\_1}{\left(\Lambda\_1 + \overline{\Lambda}\_1\right)^2} \mathbf{e}^{\eta\_1 + \overline{\eta}\_1} \tag{233}$$

where *ε<sup>i</sup>* has been absorbed into the constant *e<sup>η</sup>*<sup>10</sup> by redefiniing *η*10. In the end, we attain the one-soliton solution to the DNLS equation with VBC

$$\mu\_1(\mathbf{x}, t) = \mathbf{g}\_{\mathbf{y}} \overline{\mathbf{f}}\_1 / f\_1^2 \tag{234}$$

which is characterized with two complex parameters Λ<sup>1</sup> and *η*<sup>10</sup> and shown in **Figure 1**. If we redefine the parameter Λ<sup>1</sup> as Λ<sup>1</sup> � �i2*λ* 2 <sup>1</sup> and *λ*<sup>1</sup> � *μ*<sup>1</sup> þ i*v*1, then

$$\begin{split} \mathbf{e}^{\mathsf{i}2\hat{\mathsf{i}}\_{1}^{2}\mathbf{x} + \mathsf{i}4\hat{\mathsf{i}}\_{1}^{4}\mathbf{t} + \overline{\eta}\_{10} / 2 &\equiv \mathbf{e}^{-\Theta\_{1}}\mathbf{e}^{\mathsf{i}\Phi\_{1}}; \mathbf{e}^{\mathsf{T}\eta\_{1}}/2 \equiv \mathbf{e}^{4\mu\_{1}\mu\_{31}}\mathbf{e}^{\mathsf{i}\alpha\_{0}}; \Theta\_{1} \\ &\equiv 4\mu\_{1}\nu\_{1} \left[\mathbf{x} - \mathbf{x}\_{10} + 4\left(\mu\_{1}^{2} - \nu\_{1}^{2}\right)t\right]; \Phi\_{1} \\ &\equiv 2\left(\mu\_{1}^{2} - \nu\_{1}^{2}\right)\mathbf{x} + \left[4\left(\mu\_{1}^{2} - \nu\_{1}^{2}\right)^{2} - 16\mu\_{1}^{2}\nu\_{1}^{2}\right]t + a\_{10} \end{split} \tag{235}$$

Then

$$\begin{aligned} \mathbf{g}\_1 &= \mathbf{e}^{\eta\_1} \equiv \mathbf{e}^{-\mathbf{i}\overline{\mathbf{2}}^2 \mathbf{x} - \mathbf{i}\overline{\mathbf{4}}^4 \mathbf{i} + \eta\_{10}} = 2\mathbf{e}^{-\Theta\_1} \mathbf{e}^{-i\Phi\_1} \\\\ f\_1 &= \mathbf{1} + \frac{\mathbf{i}}{2} \frac{\Lambda\_1}{\left(\Lambda\_1 + \overline{\Lambda}\_1\right)^2} \mathbf{e}^{\eta\_1 + \overline{\eta}\_1} \\\\ &= \mathbf{1} - \frac{\overline{\lambda}^2}{\left(\lambda^2 - \overline{\lambda}^2\right)^2} \mathbf{e}^{-2\Theta} \\\\ \frac{\mathbf{g}\_1 \overline{f}\_1}{\xi^2} &= 2 \left(1 - \frac{\lambda\_1^2}{\left(\lambda\_1 - \overline{\lambda}^2\right)^2} \mathbf{e}^{-2\Theta\_1}\right) \mathbf{e}^{-\Theta\_1} \mathbf{e}^{-i\Phi\_1} \bigg/ \left(1 - \frac{\overline{\lambda}\_1^2}{\left(\lambda\_1 - \overline{\lambda}^2\right)^2} \mathbf{e}^{-2\Theta\_1}\right)^2 \end{aligned} \tag{236}$$

$$u\_1(\mathbf{x}, t) = \frac{\mathbf{g}f\_1}{f\_1^2} = 2 \left( 1 - \frac{\lambda\_1^2}{\left(\lambda\_1^2 - \overline{\lambda\_1^2}\right)^2} \mathbf{e}^{-2\Theta\_1} \right) \mathbf{e}^{-\Theta\_1} \mathbf{e}^{-i\Phi\_1} \bigg/ \left( \mathbf{1} - \frac{\lambda\_1^2}{\left(\lambda\_1^2 - \overline{\lambda\_1^2}\right)^2} \mathbf{e}^{-2\Theta\_1} \right) \tag{237}$$

It is easy to find, up to a permitted constant global phase factor e<sup>i</sup><sup>π</sup> ¼ �1, the one-soliton solution (234) or (237) gotten in this paper is in perfect agreement with that gotten from other approaches [16]. By further redefining its soliton center, initial phase and *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>*1e<sup>i</sup>*β*<sup>1</sup> , the one-soliton solution can be changed into the usual typical form [16, 23, 26, 27].

$$u\_1(\mathbf{x}, t) = 4|\lambda\_1| \sin 2\beta\_1 \left(\mathbf{1} + \mathbf{e}^{\mathrm{i}2\beta\_1} \mathbf{e}^{-2\Theta\_1}\right) \mathbf{e}^{-\Theta\_1} \mathbf{e}^{-i\Phi\_1} / \left(\mathbf{1} + \mathbf{e}^{-\mathrm{i}2\beta\_1} \mathbf{e}^{-2\Theta\_1}\right)^2 \tag{238}$$

On the other hand, just like in Ref. [13], we can rewrite *g*<sup>1</sup> and *f* <sup>1</sup> in a more appropriate or "standard" form

$$\mathbf{g\_1} = \mathbf{e^{\eta\_1 + \eta\_1}} \tag{239}$$

$$f\_1 = \mathbf{1} + \mathbf{e}^{(\eta\_1 + \rho\_1) + (\overline{\eta}\_1 + \rho\_{1'}) + \theta\_{1'}} \tag{240}$$

Here

$$\mathbf{e}^{\varrho\_1} = \mathbf{1}, \mathbf{e}^{\varrho\_{\mathbf{1'}}} = \mathbf{i}/\overline{\Lambda}\_1, \mathbf{e}^{\varrho\_{\mathbf{1}\mathbf{i'}}} = \mathbf{i}\Lambda\_1(-\overline{\mathbf{i}}\overline{\Lambda}\_1)/2\left(\Lambda\_1 + \overline{\Lambda}\_1\right)^2,\tag{241}$$

*f* <sup>2</sup> ¼ 1 þ

4 Λ<sup>1</sup> þ Λ<sup>1</sup>

iΛ<sup>1</sup> 2 Λ<sup>1</sup> þ Λ<sup>1</sup>

� <sup>Λ</sup>1Λ2j j <sup>Λ</sup><sup>1</sup> � <sup>Λ</sup><sup>2</sup> <sup>4</sup>

� �<sup>2</sup> <sup>Λ</sup><sup>2</sup> <sup>þ</sup> <sup>Λ</sup><sup>2</sup>

the two-soliton solution as

� �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*<sup>1</sup> <sup>þ</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

� �<sup>2</sup> <sup>Λ</sup><sup>1</sup> <sup>þ</sup> <sup>Λ</sup><sup>2</sup>

iΛ<sup>2</sup> 2 Λ<sup>2</sup> þ Λ<sup>2</sup>

� �<sup>2</sup> <sup>Λ</sup><sup>2</sup> <sup>þ</sup> <sup>Λ</sup><sup>1</sup>

It can also be rewritten in a standard form as follows:

<sup>þ</sup> <sup>e</sup> *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ0 *<sup>θ</sup>*12þ*θ*120þ*θ*22<sup>0</sup>

shown in **Figure 3**. By redefining parameters *η<sup>i</sup>*<sup>0</sup> and

permitted constant global phase factor.

*4.3.3 Extension to the* N-*soliton solution*

*g* ð Þ1

ð Þ1

ð Þ 0

exp <sup>X</sup><sup>2</sup>*<sup>N</sup>*

exp <sup>X</sup><sup>2</sup>*<sup>N</sup>*

<sup>0</sup> ¼ �iΛ*<sup>j</sup>* � ��<sup>1</sup>

<sup>0</sup>*k*<sup>0</sup> ¼ 2 Λ*<sup>j</sup>* � Λ*<sup>k</sup>* � �<sup>2</sup>

*gN* <sup>¼</sup> <sup>X</sup> *κ <sup>j</sup>*¼0, 1

*<sup>f</sup> <sup>N</sup>* <sup>¼</sup> <sup>X</sup> *κ <sup>j</sup>*¼0, 1

where *η<sup>N</sup>*þ*<sup>j</sup>* ¼ *η <sup>j</sup>* ð Þ *j*≤ *N*

<sup>e</sup>*<sup>ϕ</sup> <sup>j</sup>* <sup>¼</sup> 1, e*<sup>φ</sup> <sup>j</sup>*þ*<sup>N</sup>* <sup>¼</sup> <sup>e</sup>*<sup>φ</sup> <sup>j</sup>*

<sup>e</sup>*<sup>θ</sup>jk* <sup>¼</sup> <sup>2</sup> <sup>Λ</sup>*<sup>j</sup>* � <sup>Λ</sup>*<sup>k</sup>* � �<sup>2</sup>

<sup>e</sup>*<sup>θ</sup> <sup>j</sup>*þ*N*,*k*þ*<sup>N</sup>* <sup>¼</sup> <sup>e</sup>*<sup>θ</sup> <sup>j</sup>*

**55**

� �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*2þ*η*<sup>2</sup> <sup>þ</sup>

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

*<sup>g</sup>*<sup>2</sup> <sup>¼</sup> <sup>e</sup>*<sup>η</sup>*1þ*φ*<sup>1</sup> <sup>þ</sup> <sup>e</sup>*<sup>η</sup>*2þ*φ*<sup>2</sup> <sup>þ</sup> <sup>e</sup> *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ0 *<sup>θ</sup>*12þ*θ*110þ*θ*21<sup>0</sup>

*<sup>f</sup>* <sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup> *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ0 *<sup>θ</sup>*11<sup>0</sup> <sup>þ</sup> <sup>e</sup> *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ0 *<sup>θ</sup>*12<sup>0</sup> <sup>þ</sup> <sup>e</sup> *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ0 *<sup>θ</sup>*21<sup>0</sup>

Λ*<sup>k</sup>* ¼ �i2*λ*

� �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*2þ*η*1þ*η*<sup>2</sup>

<sup>þ</sup> <sup>e</sup> *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ0 *<sup>θ</sup>*22<sup>0</sup> <sup>þ</sup> <sup>e</sup> *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ *<sup>η</sup>*1þ*φ*<sup>1</sup> ð Þþ0 *<sup>η</sup>*2þ*φ*<sup>2</sup> ð Þþ0 *<sup>θ</sup>*12þ*θ*110þ*θ*120þ*θ*210þ*θ*220þ*θ*102<sup>0</sup>

where *φi*, *θij* in (249) and (250) are defined afterwards in (256). We then attain

2

<sup>2</sup> (251)

*<sup>k</sup>*, *k* ¼ 1, 2, (252)

*<sup>i</sup>* , *i* ¼ 1, 2, ⋯, *N*

1≤*j*<*k <sup>κ</sup> <sup>j</sup>κkθjk* n o (254)

1≤*j*<*k <sup>κ</sup> <sup>j</sup>κkθjk* n o (255)

� �*=*<sup>2</sup> <sup>Λ</sup>*<sup>j</sup>* <sup>þ</sup> <sup>Λ</sup>*<sup>k</sup>*

� �, 1ð Þ <sup>≤</sup>*j*<sup>≤</sup> *<sup>N</sup>*, 1<sup>≤</sup> *<sup>k</sup>*<sup>≤</sup> *<sup>N</sup>* ,

� �<sup>2</sup>

*u*2ð Þ¼ *x*, *t g*<sup>2</sup> *f* <sup>2</sup>*= f*

which is characterized with four complex parameters Λ1, Λ2, *η*10, and *η*<sup>20</sup> and

2

we can easily transform it to a two-soliton form given in Ref. [23], up to a

Generally for the case of *N*-soliton solution, if we select *g*ð Þ<sup>1</sup> in (209) to be

then using an induction method, we can write the *N*-soliton solution as

*κ <sup>j</sup> η <sup>j</sup>* þ *φ <sup>j</sup>*

*κ <sup>j</sup> η <sup>j</sup>* þ *φ <sup>j</sup>*

*<sup>=</sup>*iΛ*<sup>j</sup>* � <sup>i</sup>Λ*k*, e*<sup>θ</sup> <sup>j</sup>*,*k*þ*<sup>N</sup>* <sup>¼</sup> <sup>e</sup>*<sup>θ</sup> <sup>j</sup>*,*k*<sup>0</sup> <sup>¼</sup> <sup>i</sup>Λ*<sup>j</sup>* �iΛ*<sup>k</sup>*

*=* �iΛ*<sup>j</sup>* � � �iΛ*<sup>k</sup>*

� � þX<sup>2</sup>*<sup>N</sup>*

� � þX<sup>2</sup>*<sup>N</sup>*

*<sup>N</sup>* <sup>¼</sup> <sup>e</sup>*<sup>η</sup>*<sup>1</sup> <sup>þ</sup> <sup>e</sup>*<sup>η</sup>*<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> <sup>e</sup>*<sup>η</sup><sup>N</sup>* , *<sup>η</sup><sup>i</sup>* <sup>¼</sup> *<sup>Ω</sup>it* <sup>þ</sup> <sup>Λ</sup>*ix* <sup>þ</sup> *<sup>η</sup>*0*<sup>i</sup>*, *<sup>Ω</sup><sup>i</sup>* <sup>¼</sup> <sup>i</sup>Λ<sup>2</sup>

*j*¼1

*j*¼1

iΛ<sup>1</sup> 2 Λ<sup>1</sup> þ Λ<sup>2</sup>

� �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*<sup>2</sup> <sup>þ</sup>

iΛ<sup>2</sup> 2 Λ<sup>2</sup> þ Λ<sup>1</sup> � �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*2þ*η*<sup>1</sup>

(248)

(249)

(250)

(253)

(256)

which makes us easily extend the solution form to the case of *n*-soliton solution.

## *4.3.2 The two-soliton solution*

For the two-soliton case, again from (209), we can select *g* ð Þ1 <sup>2</sup> as

$$\mathbf{g}\_2^{(1)} = \mathbf{e}^{\eta\_1} + \mathbf{e}^{\eta\_2}, \eta\_i = \mathfrak{Q}\_i t + \Lambda\_i \mathfrak{x} + \eta\_{i0}, \mathfrak{Q}\_i = \mathbf{i} \Lambda\_i^2, i = \mathbf{1}, \mathbf{2}. \tag{242}$$

The similar procedures to that used in the one-soliton case can be used to deduce *g*<sup>2</sup> and *f* <sup>2</sup>. From (210) and (211), we can select *f* ð Þ1 <sup>2</sup> ¼ 0, then from (212), we has to select *g* ð Þ2 <sup>2</sup> ¼ 0. From (213) and (214), we can get the expressions of *f* ð Þ2 <sup>2</sup> � *f* ð Þ2 <sup>2</sup> and *f* ð Þ2 <sup>2</sup> þ *f* ð Þ2 <sup>2</sup> , then attain *f* ð Þ2 <sup>2</sup> to be

$$\begin{split} f\_{2}^{(2)} &= \frac{\mathbf{i}\Lambda\_{1}}{2\left(\Lambda\_{1} + \overline{\Lambda}\_{1}\right)^{2}} \mathbf{e}^{\eta\_{1} + \overline{\eta}\_{1}} + \frac{\mathbf{i}\Lambda\_{2}}{2\left(\Lambda\_{2} + \overline{\Lambda}\_{2}\right)^{2}} \mathbf{e}^{\eta\_{2} + \overline{\eta}\_{2}} + \frac{\mathbf{i}\Lambda\_{1}}{2\left(\Lambda\_{1} + \overline{\Lambda}\_{2}\right)^{2}} \mathbf{e}^{\eta\_{1} + \overline{\eta}\_{2}} \\ &+ \frac{\mathbf{i}\Lambda\_{2}}{2\left(\Lambda\_{2} + \overline{\Lambda}\_{1}\right)^{2}} \mathbf{e}^{\eta\_{2} + \overline{\eta}\_{1}} \end{split} \tag{243}$$

Substituting (242) and (243) into (215), one can attain *g* ð Þ3 <sup>2</sup> to be

$$\mathbf{g}\_{2}^{(3)} = \frac{-\mathbf{i}(\boldsymbol{\Lambda}\_{1} - \boldsymbol{\Lambda}\_{2})^{2}\mathbf{e}^{\eta\_{1} + \eta\_{2}}}{2} \left[ \frac{\overline{\boldsymbol{\Lambda}\_{1}}\mathbf{e}^{\eta\_{1}}}{\left(\boldsymbol{\Lambda}\_{1} + \overline{\boldsymbol{\Lambda}}\_{1}\right)^{2}\left(\boldsymbol{\Lambda}\_{2} + \overline{\boldsymbol{\Lambda}}\_{1}\right)^{2}} + \frac{\overline{\boldsymbol{\Lambda}}\_{2}\mathbf{e}^{\eta\_{2}}}{\left(\boldsymbol{\Lambda}\_{2} + \overline{\boldsymbol{\Lambda}}\_{2}\right)^{2}\left(\boldsymbol{\Lambda}\_{1} + \overline{\boldsymbol{\Lambda}}\_{2}\right)^{2}} \right] \tag{244}$$

Substituting the expressions of *g* ð Þ1 <sup>2</sup> , *g* ð Þ2 <sup>2</sup> , *g* ð Þ3 <sup>2</sup> , *f* ð Þ1 <sup>2</sup> , *f* ð Þ2 <sup>2</sup> into (216) and (217), we can select that *f* ð Þ3 <sup>2</sup> ¼ 0. Then from the expressions of *g* ð Þ1 <sup>2</sup> , *g* ð Þ2 <sup>2</sup> , *g* ð Þ3 <sup>2</sup> , *f* ð Þ1 <sup>2</sup> , *f* ð Þ2 <sup>2</sup> , *f* ð Þ3 <sup>2</sup> and (218), we can select *g* ð Þ 4 <sup>2</sup> ¼ 0. From (219) and (220), we can get the expressions of *f* ð Þ 4 <sup>2</sup> � *f* ð Þ 4 <sup>2</sup> and *f* ð Þ 4 <sup>2</sup> þ *f* ð Þ 4 <sup>2</sup> , then get *f* ð Þ 4 <sup>2</sup> to be

$$f\_{2}^{(4)} = -\frac{\Lambda\_{1}\Lambda\_{2}|\Lambda\_{1} - \Lambda\_{2}|^{4}}{4\left(\Lambda\_{1} + \overline{\Lambda}\_{1}\right)^{2}\left(\Lambda\_{2} + \overline{\Lambda}\_{2}\right)^{2}\left(\Lambda\_{1} + \overline{\Lambda}\_{2}\right)^{2}\left(\Lambda\_{2} + \overline{\Lambda}\_{1}\right)^{2}}\mathbf{e}^{\eta\_{1} + \eta\_{2} + \overline{\eta}\_{1} + \overline{\eta}\_{2}}\tag{245}$$

Due to (243) and (244), we can also easily verify that

$$(\mathrm{i}D\_t + D\_x^2)\mathrm{g}\_2^{(3)} \cdot f\_2^{(2)} = \mathbf{0} \tag{246}$$

Then from (244), (245), (246), and (221), we can select *g* ð Þ5 <sup>2</sup> ¼ 0. From (222)– (223) and so on, we find that the series of (207) and (208) can be cut off by selecting *g* ð Þ5 <sup>2</sup> , *f* ð Þ5 <sup>2</sup> ; *g* ð Þ 6 <sup>2</sup> , *f* ð Þ 6 <sup>2</sup> ⋯, all to be zero. We thus attain the last result of *g*2, *f* <sup>2</sup> to be

$$\begin{aligned} \mathfrak{g}\_2 &= \mathbf{e}^{\eta\_1} + \mathbf{e}^{\eta\_2} \\ &- \frac{\mathbf{i} \left(\Lambda\_1 - \Lambda\_2\right)^2}{2} \mathbf{e}^{\eta\_1 + \eta\_2} \mathbf{e} \left[ \frac{\overline{\Lambda}\_1}{\left(\Lambda\_1 + \overline{\Lambda}\_1\right)^2 \left(\Lambda\_2 + \overline{\Lambda}\_1\right)^2} \mathbf{e}^{\overline{\eta}\_1} + \frac{\overline{\Lambda}\_2}{\left(\Lambda\_2 + \overline{\Lambda}\_2\right)^2 \left(\Lambda\_1 + \overline{\Lambda}\_2\right)^2} \mathbf{e}^{\overline{\eta}\_2} \right] \end{aligned} \tag{247}$$

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

*f* <sup>2</sup> ¼ 1 þ iΛ<sup>1</sup> 2 Λ<sup>1</sup> þ Λ<sup>1</sup> � �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*<sup>1</sup> <sup>þ</sup> iΛ<sup>2</sup> 2 Λ<sup>2</sup> þ Λ<sup>2</sup> � �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*2þ*η*<sup>2</sup> <sup>þ</sup> iΛ<sup>1</sup> 2 Λ<sup>1</sup> þ Λ<sup>2</sup> � �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*<sup>2</sup> <sup>þ</sup> iΛ<sup>2</sup> 2 Λ<sup>2</sup> þ Λ<sup>1</sup> � �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*2þ*η*<sup>1</sup> � <sup>Λ</sup>1Λ2j j <sup>Λ</sup><sup>1</sup> � <sup>Λ</sup><sup>2</sup> <sup>4</sup> 4 Λ<sup>1</sup> þ Λ<sup>1</sup> � �<sup>2</sup> <sup>Λ</sup><sup>2</sup> <sup>þ</sup> <sup>Λ</sup><sup>2</sup> � �<sup>2</sup> <sup>Λ</sup><sup>1</sup> <sup>þ</sup> <sup>Λ</sup><sup>2</sup> � �<sup>2</sup> <sup>Λ</sup><sup>2</sup> <sup>þ</sup> <sup>Λ</sup><sup>1</sup> � �<sup>2</sup> <sup>e</sup>*<sup>η</sup>*1þ*η*2þ*η*1þ*η*<sup>2</sup> (248)

It can also be rewritten in a standard form as follows:

$$\begin{aligned} \mathbf{g}\_2 &= \mathbf{e}^{\eta\_1 + \rho\_1} + \mathbf{e}^{\eta\_2 + \rho\_2} + \mathbf{e}^{(\eta\_1 + \rho\_1) + (\eta\_2 + \rho\_2) + (\overline{\eta}\_1 + \rho\_{1'}) + \theta\_{2'} + \theta\_{2'} + \theta\_{2'}} \\ &+ \mathbf{e}^{(\eta\_1 + \rho\_1) + (\eta\_2 + \rho\_2) + (\overline{\eta}\_2 + \rho\_{2'}) + \theta\_{21'} + \theta\_{2'} + \theta\_{2'}} \end{aligned} \tag{249}$$
 
$$\begin{aligned} \mathbf{f}\_2 &= \mathbf{1} + \mathbf{e}^{(\eta\_1 + \eta\_1) + (\overline{\eta}\_1 + \eta\_{1'}) + \theta\_{11'}} + \mathbf{e}^{(\eta\_1 + \eta\_1) + (\overline{\eta}\_2 + \rho\_{2'}) + \theta\_{12'}} + \mathbf{e}^{(\eta\_2 + \eta\_2) + (\overline{\eta}\_1 + \eta\_{1'}) + \theta\_{21'}} \\ &+ \mathbf{e}^{(\eta\_2 + \rho\_2) + (\overline{\eta}\_2 + \rho\_{2'}) + \theta\_{22'}} + \mathbf{e}^{(\eta\_1 + \eta\_1) + (\eta\_2 + \rho\_2) + (\overline{\eta}\_1 + \rho\_{2'}) + \theta\_{21'} + \theta\_{21'} + \theta\_{22'} + \theta\_{12'}} + \mathbf{e}^{(\eta\_2 + \eta\_2) + (\overline{\eta}\_{2'} + \eta\_{2'}) + \theta\_{22'}} \end{aligned} \tag{250}$$

where *φi*, *θij* in (249) and (250) are defined afterwards in (256). We then attain the two-soliton solution as

$$
\mu\_2(\mathbf{x}, t) = \mathbf{g}\_2 \overline{f}\_2 / f\_2^2 \tag{251}
$$

which is characterized with four complex parameters Λ1, Λ2, *η*10, and *η*<sup>20</sup> and shown in **Figure 3**. By redefining parameters *η<sup>i</sup>*<sup>0</sup> and

$$
\Lambda\_k = -\text{i}\mathbf{2}\overline{\lambda}\_k^2, k = \mathbf{1}, \mathbf{2}, \tag{252}
$$

we can easily transform it to a two-soliton form given in Ref. [23], up to a permitted constant global phase factor.

#### *4.3.3 Extension to the* N-*soliton solution*

Generally for the case of *N*-soliton solution, if we select *g*ð Þ<sup>1</sup> in (209) to be

$$\begin{aligned} \mathbf{g}\_{N}^{(1)} &= \mathbf{e}^{\eta\_{1}} + \mathbf{e}^{\eta\_{2}} + \dots + \mathbf{e}^{\eta\_{N}}, \\ \eta\_{i} &= \Omega\_{i}t + \Lambda\_{i}\mathbf{x} + \eta\_{0i}, \Omega\_{i} = \mathbf{i}\Lambda\_{i}^{2}, i = \mathbf{1}, 2, \dots, N \end{aligned} \tag{253}$$

then using an induction method, we can write the *N*-soliton solution as

$$\mathbf{g}\_N = \sum\_{\kappa\_j=0,1} {\text{2}\mathbf{1}}^{(1)} \exp\left\{ \sum\_{j=1}^{2N} \kappa\_j \left(\eta\_j + \varrho\_j\right) + \sum\_{1 \le j < k}^{2N} \kappa\_j \kappa\_k \theta\_{jk} \right\} \tag{254}$$

$$f\_N = \sum\_{\kappa\_j=0,1} {^{(0)}} \exp\left\{ \sum\_{j=1}^{2N} \kappa\_j \left( \eta\_j + \varrho\_j \right) + \sum\_{1 \le j < k}^{2N} \kappa\_j \kappa\_k \theta\_{jk} \right\} \tag{255}$$

where *η<sup>N</sup>*þ*<sup>j</sup>* ¼ *η <sup>j</sup>* ð Þ *j*≤ *N*

$$\begin{split} \mathbf{e}^{\boldsymbol{\theta}\_{j}} &= \mathbf{1}, \mathbf{e}^{\boldsymbol{\theta}\_{j+N}} = \mathbf{e}^{\boldsymbol{\theta}\_{j'}} = \left(-\mathbf{i}\overline{\boldsymbol{\Lambda}}\_{j}\right)^{-1} \\ \mathbf{e}^{\boldsymbol{\theta}\_{k}} &= 2\left(\boldsymbol{\Lambda}\_{j} - \boldsymbol{\Lambda}\_{k}\right)^{2} / \mathbf{i}\boldsymbol{\Lambda}\_{j} \cdot \mathbf{i}\boldsymbol{\Lambda}\_{k}, \mathbf{e}^{\boldsymbol{\theta}\_{j'},k+N} = \mathbf{e}^{\boldsymbol{\theta}\_{j,k'}} = \mathbf{i}\boldsymbol{\Lambda}\_{j} \left(-\mathbf{i}\overline{\boldsymbol{\Lambda}}\_{k}\right) / 2\left(\boldsymbol{\Lambda}\_{j} + \overline{\boldsymbol{\Lambda}}\_{k}\right)^{2} \\ \mathbf{e}^{\boldsymbol{\theta}\_{j+N,k+N}} &= \mathbf{e}^{\boldsymbol{\theta}\_{j'}} = 2\left(\overline{\boldsymbol{\Lambda}}\_{j} - \overline{\boldsymbol{\Lambda}}\_{k}\right)^{2} / \left(-\mathbf{i}\overline{\boldsymbol{\Lambda}}\_{j}\right) \left(-\mathbf{i}\overline{\boldsymbol{\Lambda}}\_{k}\right), \left(\mathbb{1} \leq j \leq N, \mathbf{1} \leq k \leq N\right), \end{split} \tag{256}$$

therein P *κ <sup>j</sup>*¼0,1 ð Þ*<sup>l</sup>* represents a summation over *<sup>κ</sup> <sup>j</sup>* <sup>¼</sup> 0, 1 under the condition P*<sup>N</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>κ</sup> <sup>j</sup>* <sup>¼</sup> *<sup>l</sup>* <sup>þ</sup> <sup>P</sup>*<sup>N</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>κ</sup> <sup>j</sup>*þ*<sup>N</sup>*.

Here, we have some discussion in order. Because what concerns us only is the soliton solutions, our soliton solution of DNLS equation with VBC is only a subset of the whole solution set. Actually in the whole process of deriving the bilinear-form equations and searching for the one and two-soliton solutions, some of the latter results are only the sufficient but not the necessary conditions of the former equations. Thereby some possible modes might have been missing. For example, the solutions of Eqs. (209)–(211) are not as unique as in (224) and (225), some other possibilities thus get lost here. This is also why we use a term "select" to determine a solution of an equation. In another word, we have selected a soliton solution. Meanwhile, we have demonstrated in **Figures 2** and **3**, the three-dimensional evolution of the one- and two-soliton amplitude with time and space, respectively. The elastic collision of two solitons in the two-soliton case has been demonstrated in **Figure 4(a**–**d)** too. It can be found that each soliton keeps the same form and characteristic after the collision as that before the collision. In this section, by means of introducing HBDT and employing an appropriate solution form (192),

we successfully solve the derivative nonlinear Schrödinger equation with VBC. The one- and two-soliton solutions are derived and their equivalence to the existing results is manifested. The *N*-soliton solution has been given by an induction

*The elastic collision between two solitons at 4 typical moments: (a) t = 10(normalized time); (b) t = 1;*

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1*

*(c) t = 1; (d) t = 10, from 10 before collision to 10 after collision.*

*DOI: http://dx.doi.org/10.5772/intechopen.93438*

method. On the other hand, by using simple parameter transformations (e.g., (235) and (252)), the soliton solutions attained here can be changed into or equivalent to that gotten based on IST, up to a permitted global constant phase factor. This section impresses us so greatly for a fact that, ranked with the extensively used IST [23] and other methods, the HBDT is another effective and important tool to deal with a partial differential equation. It is especially suitable for some integrable

Department of Physics, Wuhan University, Wuhan, P.R. China

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: zgq@whu.edu.cn

provided the original work is properly cited.

nonlinear models.

**Figure 4.**

**Author details**

Zhou Guo-Quan

**57**

**Figure 2.** *The evolution of one-soliton solution with time and space under parameter* Λ<sup>1</sup> ¼ �1 þ 0*:*2i, *η*<sup>10</sup> ¼ 1 *in (234).*

#### **Figure 3.**

*The evolution of two-soliton solution with time and space under parameter* Λ<sup>1</sup> ¼ 1 þ 0*:*3i, Λ<sup>2</sup> ¼ 1 � 0*:*3i, *η*<sup>10</sup> ¼ *η*<sup>20</sup> ¼ 1 *in (251).*

*Soliton and Rogue-Wave Solutions of Derivative Nonlinear Schrödinger Equation - Part 1 DOI: http://dx.doi.org/10.5772/intechopen.93438*

#### **Figure 4.**

*The elastic collision between two solitons at 4 typical moments: (a) t = 10(normalized time); (b) t = 1; (c) t = 1; (d) t = 10, from 10 before collision to 10 after collision.*

we successfully solve the derivative nonlinear Schrödinger equation with VBC. The one- and two-soliton solutions are derived and their equivalence to the existing results is manifested. The *N*-soliton solution has been given by an induction method. On the other hand, by using simple parameter transformations (e.g., (235) and (252)), the soliton solutions attained here can be changed into or equivalent to that gotten based on IST, up to a permitted global constant phase factor. This section impresses us so greatly for a fact that, ranked with the extensively used IST [23] and other methods, the HBDT is another effective and important tool to deal with a partial differential equation. It is especially suitable for some integrable nonlinear models.

## **Author details**

Zhou Guo-Quan Department of Physics, Wuhan University, Wuhan, P.R. China

\*Address all correspondence to: zgq@whu.edu.cn

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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**Chapter 3**

**Abstract**

periodic solution.

**61**

*Zhou Guo-Quan*

Soliton and Rogue-Wave Solutions

A revised and rigorously proved inverse scattering transform (IST for brevity) for DNLS+ equation, with a constant nonvanishing boundary condition (NVBC) and normal group velocity dispersion, is proposed by introducing a suitable affine parameter in the Zakharov-Shabat IST integral; the explicit breather-type and pure *N*-soliton solutions had been derived by some algebra techniques. On the other hand, DNLS equation with a non-vanishing background of harmonic plane wave is also solved by means of Hirota's bilinear formalism. Its space periodic solutions are determined, and its rogue wave solution is derived as a long-wave limit of this space

**Keywords:** soliton, nonlinear equation, derivative nonlinear Schrödinger equation, inverse scattering transform, Zakharov-Shabat equation, Hirota's bilinear derivative

**1. Breather-type and pure** *N***-soliton solution to DNLS+ equation with**

DNLS+ equation with NVBC, the concerned theme of this section, is only a transformed version of modified nonlinear Schrödinger equation with normal group velocity dispersion and a nonlinear self-steepen term and can be expressed as

> *u*

Some progress have been made by several researchers to solve the DNLS equation for DNLS equation with NVBC, many heuristic and interesting results have been attained [1–14]. An early proposed IST worked on the Riemann sheets can only determine the modulus of the one-soliton solution [3, 15]. References [4, 5, 16] had attained a pure single dark/bright soliton solution. Reference [6] had derived a formula for *N*� soliton solution in terms of Vandermonde-like determinants by means of Bäcklund transformation; but just as reference [7, 9] pointed out, this multi-soliton solution is still difficult to exhibit the internal elastic collisions among solitons and the typical asymptotic behaviors of multi-soliton of DNLS equation. By introducing an affine parameter in the integral of Zakharov-Shabat IST, reference [7] had found their pure *N*� soliton solution for a special case that all the simple

*<sup>x</sup>* <sup>¼</sup> 0, (1)

*iut* � *uxx* <sup>þ</sup> *i u*j j<sup>2</sup>

method, DNLS equation, space periodic solution, rogue wave

here the subscripts represent partial derivatives.

**NVBC based on revised IST**

of Derivative Nonlinear

Schrödinger Equation - Part 2

## **Chapter 3**
