2. Nonlinear complex exponential functions

The definitions of the functions and their properties are similar to those used in Jacobi's elliptic functions [1, 2, 4]. Let us start with the definition of our complex exponential nonlinear functions:

$$\mathbf{cnc}(u,a) = a \;e^{i\mathbf{x}} + b \;e^{-i\mathbf{x}}, \qquad \mathbf{snc}(u,a) = a \;e^{i\mathbf{x}} - b \;e^{-i\mathbf{x}}, \tag{1}$$

$$\text{dnc}(u, a) = \sqrt{1 - a \left| \text{cnc}(u) \right|^2}, \qquad \text{ncc}(u, a) = \frac{1}{\text{cnc}(u, a)}, \tag{2}$$

$$\mathsf{nsc}(u,a) = \frac{1}{\mathsf{snc}(u,a)}, \qquad \qquad \mathsf{ndc}(u,a) = \frac{1}{\mathsf{dnc}(u,a)}, \tag{3}$$

$$\mathsf{csc}(u,a) = \frac{\mathsf{sinc}(u,a)}{\mathsf{cinc}(u,a)}, \qquad \qquad \qquad \mathsf{coc}(u,a) = \frac{\mathsf{cinc}(u,a)}{\mathsf{sinc}(u,a)}.\tag{4}$$

where <sup>α</sup>, a, b∈R, and they are such that <sup>α</sup> < 1=max ð Þ <sup>a</sup> � <sup>b</sup> <sup>2</sup> h i. With these choices, the function dnc is always positive, and we do not have to worry about branch points in the relation between the variables x and u. The variables u and x are related as

$$\mu = \int\_0^\infty \frac{dt}{\sqrt{1 - a|\mathbf{cnc}(t, a)|^2}}. \tag{5}$$

A plot of these functions is found in Figure 1 for a particular set of values of the parameters. These functions behave like the usual superposition of complex exponential functions (α ¼ 0), changing behavior as the value of α increases until

## Figure 1.

Nonlinear complex exponential functions with a ¼ 0:1, b ¼ 0:9, and α ¼ 0:9. The curves correspond to 1, j j <sup>c</sup>nc uð Þ ; <sup>α</sup> <sup>2</sup> ; 2, j j <sup>s</sup>nc uð Þ ; <sup>α</sup> <sup>2</sup> ; and 3, dnc uð Þ ; α .

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential DOI: http://dx.doi.org/10.5772/intechopen.80938

it reaches the soliton value, <sup>α</sup> <sup>¼</sup> <sup>1</sup>=max ð Þ <sup>a</sup> � <sup>b</sup> <sup>2</sup> h i. The functions become concentrated around the origin for the soliton value of α.

The quarter period of these functions is defined as

$$Kc = \int\_0^{\pi/2} \frac{dt}{\sqrt{1 - a|\mathbf{cnc}(t, a)|^2}}. \tag{6}$$

If we call

The functions we are about to introduce can be used, for instance, in the case of a free Bose-Einstein condensate reflected by a potential barrier. One might be able to further analyze nonlinear tunneling [7] and nonlinear optics phenomena with the

The definitions of the functions and their properties are similar to those used in Jacobi's elliptic functions [1, 2, 4]. Let us start with the definition of our complex

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � αj j cncð Þ u

where <sup>α</sup>, a, b∈R, and they are such that <sup>α</sup> < 1=max ð Þ <sup>a</sup> � <sup>b</sup> <sup>2</sup> h i

u ¼ ðx 0

the function dnc is always positive, and we do not have to worry about branch points in the relation between the variables x and u. The variables u and x are

cncð Þ¼ <sup>u</sup>; <sup>α</sup> a eix <sup>þ</sup> b e�ix, sncð Þ¼ <sup>u</sup>; <sup>α</sup> a eix � b e�ix, (1)

sncð Þ <sup>u</sup>; <sup>α</sup> , ndcð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>1</sup>

dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>α</sup>j j cncð Þ <sup>t</sup>; <sup>α</sup> <sup>2</sup>

A plot of these functions is found in Figure 1 for a particular set of values of the parameters. These functions behave like the usual superposition of complex exponential functions (α ¼ 0), changing behavior as the value of α increases until

Nonlinear complex exponential functions with a ¼ 0:1, b ¼ 0:9, and α ¼ 0:9. The curves correspond to 1,

cncð Þ <sup>u</sup>; <sup>α</sup> , cocð Þ¼ <sup>u</sup>; <sup>α</sup> cncð Þ <sup>u</sup>; <sup>α</sup>

, nccð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>1</sup>

<sup>q</sup> : (5)

cncð Þ <sup>u</sup>; <sup>α</sup> , (2)

dncð Þ <sup>u</sup>; <sup>α</sup> , (3)

sncð Þ <sup>u</sup>; <sup>α</sup> : (4)

. With these choices,

help of these functions.

related as

Figure 1.

j j <sup>c</sup>nc uð Þ ; <sup>α</sup> <sup>2</sup>

36

; 2, j j <sup>s</sup>nc uð Þ ; <sup>α</sup> <sup>2</sup>

; and 3, dnc uð Þ ; α .

exponential nonlinear functions:

dncð Þ¼ u; α

nscð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>1</sup>

tacð Þ¼ <sup>u</sup>; <sup>α</sup> sncð Þ <sup>u</sup>; <sup>α</sup>

2. Nonlinear complex exponential functions

Nonlinear Optics ‐ Novel Results in Theory and Applications

q

$$n\_0 = a^2 + b^2, \qquad \qquad n\_1 = 1 - 2a(a^2 + b^2), \tag{7}$$

$$n\_2 = 1 - \frac{3}{2}a(a^2 + b^2), \qquad n\_3 = 1 - a(a^2 + b^2), \tag{8}$$

$$n\_4 = 1 - \frac{a}{2}(a^2 + b^2), \qquad n\_5 = 1 + \frac{a}{2}(a^2 + b^2), \tag{9}$$

$$n\_6 = \mathbf{1} + a(a^2 + b^2), \qquad \qquad n\_7 = \mathbf{1} + \frac{3}{2}a(a^2 + b^2), \tag{10}$$

the squares of the nonlinear functions are written as

$$\text{cnc}^2(u, a) - \text{snc}^2(u, a) = 4ab,\tag{11}$$

$$\left|\mathsf{cnc}(\mathfrak{u},a)\right|^2 + \left|\mathsf{snc}(\mathfrak{u},a)\right|^2 = 2n\_0 \tag{12}$$

$$\text{dnc}^2(u, a) = \mathbf{1} - a|\text{cnc}(u, a)|^2 \tag{13}$$

$$\mathbf{u} = \mathbf{u}\_1 + a \left| \text{sinc}(\boldsymbol{u}, \boldsymbol{a}) \right|^2,\tag{14}$$

$$\text{trace}^2(u, a) = 1 - 4ab \text{ ncc}^2(u, a), \tag{15}$$

$$\mathbf{c}\mathbf{c}\mathbf{c}^{2}(u,a) = \mathbf{1} + 4ab\,\mathbf{n}\mathbf{c}\mathbf{c}^{2}(u,a). \tag{16}$$

Some derivatives of these functions are

$$\mathsf{ncnc}'(u,a) = i \; \mathsf{snc}(u,a) \; \mathsf{dnc}(u,a), \tag{17}$$

$$\mathsf{snc}'(u,a) = i \colon \mathsf{cnc}(u,a) \cdot \mathsf{dnc}(u,a),\tag{18}$$

$$\text{clnc}'(u, a) = a \mathfrak{F}\{\text{cnc}^\*(u, a) \mid \text{snc}(u, a)\}, \tag{19}$$

$$\mathsf{ncc}'(u,a) = -i\mathsf{tac}(u,a) \cdot \mathsf{ncc}(u,a) \cdot \mathsf{dnc}(u,a),\tag{20}$$

$$\operatorname{nsc}'(u,a) = -i\operatorname{coc}(u,a) \cdot \operatorname{nsc}(u,a) \cdot \operatorname{dnc}(u,a),\tag{21}$$

$$\text{ndc}^{\prime}(u,a) = -a \mid \text{ndc}^{2}(u,a) \; \mathfrak{F}\{\text{cnc}^{\*}(u,a) \; \text{snc}(u,a)\},\tag{22}$$

$$\mathsf{tac}'(u,a) = i\left[\mathbf{1} + \mathsf{tac}^2(u,a)\right] \mathsf{d}\mathsf{nc}(u,a) \tag{23}$$

$$\mathsf{acc}'(u,a) = -i4ab\,\mathsf{nsc}^2(u,a)\mathsf{dnc}(u,a),\tag{24}$$

where ℑ indicates to take the imaginary part of the quantity. We also have that the derivative of the inverse functions is given by

$$\frac{d}{dy}\text{cnc}^{-1}(y) = \pm \frac{i}{\sqrt{\left(y^2 - 4ab\right)\left(1 - a|y|^2\right)}},\tag{25}$$

$$\frac{d}{dy}\text{snc}^{-1}(y) = \pm \frac{i}{\sqrt{(4ab+y^2)\left(n\_1+a|y|^2\right)}},\tag{26}$$

The differential equations for cnc ð Þ u; α and snc ð Þ u; α would have the Gross-Pitaevskii equation form if any of α, a, or b becomes zero or when a ¼ b (which is the case of real functions, i.e., Jacobi's functions). The case of α, a, or b zero corresponds to the cases when there is no nonlinear interaction or when there is

A straight forward application of the functions introduced in this section is the finding of the eigenfunctions of the Gross-Pitaevskii equation for a step potential:

V uð Þ¼ <sup>0</sup>, when <sup>u</sup> < 0,

and a chemical potential μ larger than the potential height V0. The Gross-

2ML<sup>2</sup> ћ2

where ψð Þ u is the unnormalized eigenfunction for the Bose-Einstein condensate (BEC), M is the mass of a single atom, N is the number of atoms in the condensate,

is a scaling length, A is the integral of the magnitude squared of the wave function, u is a dimensionless length, μ is the chemical potential, and V<sup>0</sup> is an external

For u < 0 (we call it the region I, V<sup>0</sup> ¼ 0), we use the cnc function with a ¼ 1, i.e.,

μ

NU<sup>0</sup>

a=M characterizes the atom-atom interaction, a is the scattering length, L

<sup>A</sup><sup>2</sup> NU0j j <sup>ψ</sup>ð Þ <sup>u</sup>

2

ψIð Þ¼ u; α cncð Þ kIu; α<sup>I</sup> , (41)

<sup>ћ</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>I</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> , (42)

<sup>2</sup>μA<sup>2</sup> � NU<sup>0</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> , (44)

ψIIð Þ¼ u cncð Þ kIIu; αII , (46)

<sup>2</sup>A<sup>2</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> : (45)

ψð Þ¼ u 0, (40)

V0, when u≥0,

(39)

(43)

total reflection or only transmission in a quantum system.

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

<sup>ћ</sup><sup>2</sup> ð Þ <sup>μ</sup> � <sup>V</sup><sup>0</sup> <sup>ψ</sup>ð Þ� <sup>u</sup>

k2

<sup>I</sup> <sup>¼</sup> <sup>2</sup>ML<sup>2</sup>

<sup>α</sup><sup>I</sup> <sup>¼</sup> ML<sup>2</sup>

<sup>α</sup><sup>I</sup> <sup>¼</sup> NU<sup>0</sup>

k2 I <sup>2</sup>ML<sup>2</sup> <sup>þ</sup>

<sup>μ</sup> <sup>¼</sup> <sup>ћ</sup><sup>2</sup>

For u>0, we use the nonlinear plane wave (a ¼ T, b ¼ 0)

ћ2 A2 k2 I

NU<sup>0</sup>

This last result for μ is in agreement with the conjecture formulated by D'Agosta et al. in Ref. [9], with the last term being the self-energy of the condensate, which is

2.1 The potential step

Pitaevskii equation is written as

2ML<sup>2</sup>

DOI: http://dx.doi.org/10.5772/intechopen.80938

From these equations, we obtain

d2 ψð Þ u du<sup>2</sup> <sup>þ</sup>

constant potential.

with parameters

<sup>U</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup>πћ<sup>2</sup>

and

39

independent of kI.

$$\frac{d}{dy}\text{ncc}^{-1}(y) = \pm \frac{i}{\mathcal{Y}\sqrt{(1 - 4ab\,\mathcal{Y}^2)\left(1 - a/|y|^2\right)}},\tag{27}$$

$$\frac{d}{dy}\text{nsc}^{-1}(y) = \pm \frac{i}{\mathcal{Y}\sqrt{(1+4ab\,\mathcal{Y}^2)\left(n\_1+a/|y|^2\right)}}\,\text{}\tag{28}$$

Now, the second derivatives are as follows

$$\mathsf{cnc}^{\prime\prime}(u,a) = \left[2a \mid \mathsf{cnc}(u,a)\right]^2 - n\_6 \mathsf{cnc}(u,a) - 2aab \; \mathsf{cnc}^\*(u,a),\tag{29}$$

$$\begin{split} \mathsf{snc}''(u,a) &= \left( \left[ \mathsf{3a}(a^2+b^2) - \mathsf{1} \right] - 2a \mathsf{snc}(u,a) \right]^2 \\ &\qquad \mathsf{snc}(u,a) - 2aab \ \mathsf{snc}^\*(u,a), \end{split} \tag{30}$$

$$\mathsf{Indc}''(u,a) = \mathsf{Z}\left[\mathsf{Indc}^2(u,a) - an\_0\right]\mathsf{Indc}(u,a),\tag{31}$$

$$\mathsf{ncc}^{\prime\prime}(u,a) = n\_{\mathsf{6}}\mathsf{ncc}(u,a) - 2a\mathsf{ncc}^\*(u,a) + 2aab\mathsf{ncc}^2(u,a)\mathsf{ncc}^\*(u,a),\tag{32}$$

$$\begin{split} \mathsf{nsc}\prime(u,a) &= (n\_3 + 8abn\_1 \mathsf{nsc}^2(u,a)) \mathsf{nsc}(u,a) \\ &- a[1 + 10ab \,\mathsf{nsc}^2(u,a)] \mathsf{nsc}^\*(u,a), \end{split} \tag{33}$$

$$\begin{split} \mathsf{ndc}^{\prime\prime}(\boldsymbol{u},a) &= 2a^2 \mathsf{ndc}^3(\boldsymbol{u},a) \left( \mathsf{3cnc}(\boldsymbol{u},a) \mathsf{ncc}(\boldsymbol{u},a) \right)^2 \\ &+ 2a(a^2+b^2) \mathsf{ndc}(\boldsymbol{u},a), \end{split} \tag{34}$$

$$\begin{aligned} \mathsf{tac}''(u,a) &= [1 + \mathsf{tac}^2(u,a)] \\ &\quad \left\{ a[2\mathsf{tac}(u,a) + i\,\mathfrak{T}\mathsf{tac}(u,a)] |\mathsf{cnc}(u,a)|^2 - 2\mathsf{tac}(u,a) \right\}, \end{aligned} \tag{35}$$

$$\mathsf{ccc}''(u,a) = 2[1 - \mathsf{ccc}^2(u,a)]\mathsf{ccc}(u,a)\mathsf{d}\mathsf{n}^2(u,a)$$

$$\begin{aligned} \text{vec}(\cdots,\cdots) &= \text{cs}(\cdots(\cdots)) \cdot \text{sinc}(\cdots,\cdots) \cdot \text{sinc}(\cdots,\cdots) \\ &- 2aab \left[ \frac{\text{cinc}^\*(u,a)}{\text{cinc}(u,a)} - \frac{\text{sinc}^\*(u,a)}{\text{sinc}(u,a)} \right] \text{cinc}(u,a) . \end{aligned} \tag{36}$$

The first three of the above equations can be thought of as modifications of the Gross-Pitaevskii equation, which allows for solutions of the form cnc ð Þ u; α , snc ð Þ u; α , and dnc ð Þ u; α . However, when a or b vanishes, we get the Gross-Pitaevskii form.

With these results at hand, we can see that the probability current densities associated with cnc ð Þ u; α and snc ð Þ u; α are given by

$$\begin{split} j\_c(u) &= \text{Re}\left\{ \text{cnc}^\*(u, a) \left[ -i \frac{d}{du} \text{cnc}(u, a) \right] \right\} \\ &= (a^2 - b^2) \text{ \text{dnc}(u, a)}, \end{split} \tag{37}$$

$$\begin{split} j\_s(u) &= \text{Re}\left\{ \text{snc}^\*(u, a) \left[ -i \frac{d}{du} \text{snc}(u, a) \right] \right\} \\ &= (a^2 - b^2) \text{ \text{dnc}(u, a)}, \end{split} \tag{38}$$

respectively. The nonlinear term causes that the quantum flux be no longer constant (as is the case for linear interaction) but modulated by dnc ð Þ u; α instead. Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential DOI: http://dx.doi.org/10.5772/intechopen.80938

The differential equations for cnc ð Þ u; α and snc ð Þ u; α would have the Gross-Pitaevskii equation form if any of α, a, or b becomes zero or when a ¼ b (which is the case of real functions, i.e., Jacobi's functions). The case of α, a, or b zero corresponds to the cases when there is no nonlinear interaction or when there is total reflection or only transmission in a quantum system.
