5. Remarks

Thus, we were able to obtain three sets of nonlinear functions which are solutions to the Gross-Pitaevskii equation. With these functions, we have the nonlinear versions of the trigonometric, real, and complex exponential functions and their linear combinations, and a complete set of functions as in the linear counterpart.

Due to the method of solution, which makes use of elliptic functions, these functions will expand the set of solutions that can be given to polynomial nonlinear equations, in general [8, 12–25].

For instance, a well-known optical phenomenon is the nonlinear dispersion in parabolic law medium with Kerr law nonlinearity [24]. This system is described by a nonlinear Schrödinger equation:

$$a\Psi\_t + a\Psi\_{\text{xx}} + b\left|\Psi\right|^2\Psi + c\left|\Psi\right|^4\Psi + d\left(\left|\Psi\right|^2\right)\_{\text{xx}}\Psi = \mathbf{0},\tag{142}$$

where a subindex indicates a derivative with respect to that index. The second term of the above equation represents the group velocity dispersion, the third and fourth terms are the parabolic law nonlinearity, and the last term is the nonlinear dispersion. Some solutions of Eq. (142) were found in Ref. [24]. A solution is the traveling wave, with Jacobi's sn function profile, given by

$$\Psi(\varkappa, t) = A \operatorname{sn} [B(\varkappa - \nu t), m] e^{i\phi},\tag{143}$$

$$B = \left(\frac{-bA^2}{am(1+m) - 2d(m^2+m+2)A^2}\right)^{1/2},\tag{144}$$

$$
\omega = B^2 \left( 2dA^2 - a(1+m) \right). \tag{145}
$$

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

where v ¼ �2ak is the velocity, k is the soliton frequency, ω is the soliton wave number, θ is the phase constant, and 0 < m < 1 is the modulus of Jacobi's elliptic function.

A second solution was given as

<sup>u</sup> <sup>¼</sup> ffiffiffiffiffiffiffi <sup>2</sup>ab <sup>p</sup> <sup>ð</sup><sup>x</sup>

> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffi <sup>2</sup>ab <sup>p</sup> ab

> > � e

�

with complex argument:

functions.

5. Remarks

counterpart.

48

equations, in general [8, 12–25].

a nonlinear Schrödinger equation:

<sup>i</sup>Ψ<sup>t</sup> <sup>þ</sup> <sup>a</sup>Ψxx <sup>þ</sup> <sup>b</sup>j j <sup>Ψ</sup> <sup>2</sup>

traveling wave, with Jacobi's sn function profile, given by

<sup>B</sup> <sup>¼</sup> �bA<sup>2</sup>

0

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>p</sup>

p

dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>a</sup>2e2<sup>t</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup>

Nonlinear Optics ‐ Novel Results in Theory and Applications

�2<sup>x</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2

where <sup>2</sup>F<sup>1</sup> is the hypergeometric function.

where F is elliptic integral of the first kind.

u ¼ ðx 0

e�2<sup>x</sup> þ a2e2<sup>x</sup>

<sup>b</sup><sup>2</sup> � <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �

e�2<sup>t</sup>

<sup>b</sup><sup>2</sup> � <sup>e</sup>

2F1 3 <sup>4</sup> ; <sup>1</sup>; 1 <sup>4</sup> ; � <sup>a</sup><sup>2</sup> b2

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>4</sup><sup>m</sup> sin h2

� � ���

When a ¼ b ¼ 1, the nonlinear functions reduce to Jacobi's elliptic functions

ð Þt

This is the minimum set of properties of the exponential-type nonlinear

Thus, we were able to obtain three sets of nonlinear functions which are solutions to the Gross-Pitaevskii equation. With these functions, we have the nonlinear versions of the trigonometric, real, and complex exponential functions and their linear combinations, and a complete set of functions as in the linear

Due to the method of solution, which makes use of elliptic functions, these functions will expand the set of solutions that can be given to polynomial nonlinear

For instance, a well-known optical phenomenon is the nonlinear dispersion in parabolic law medium with Kerr law nonlinearity [24]. This system is described by

<sup>Ψ</sup> <sup>þ</sup> <sup>c</sup>j j <sup>Ψ</sup> <sup>4</sup>

Ψð Þ¼ x; t A sn½ � B xð Þ � vt ; m e

amð Þ� <sup>1</sup> <sup>þ</sup> <sup>m</sup> <sup>2</sup>d mð Þ <sup>2</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup> <sup>2</sup> <sup>A</sup><sup>2</sup>

!<sup>1</sup>=<sup>2</sup>

where a subindex indicates a derivative with respect to that index. The second term of the above equation represents the group velocity dispersion, the third and fourth terms are the parabolic law nonlinearity, and the last term is the nonlinear dispersion. Some solutions of Eq. (142) were found in Ref. [24]. A solution is the

<sup>Ψ</sup> <sup>þ</sup> <sup>d</sup> j j <sup>Ψ</sup> <sup>2</sup> � �

<sup>ω</sup> <sup>¼</sup> <sup>B</sup><sup>2</sup> <sup>2</sup>dA<sup>2</sup> � <sup>a</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>m</sup> � �: (145)

<sup>q</sup> dt ¼ �iF ix ð Þ <sup>j</sup>4<sup>m</sup> , (141)

<sup>2</sup><sup>x</sup> b<sup>2</sup> e �2<sup>x</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> e 2x � �

� � � � �

2F1 3 <sup>4</sup> ; <sup>1</sup>; 1 <sup>4</sup> ; � <sup>a</sup>2e4<sup>x</sup> b2

, (140)

xx<sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>, (142)

<sup>i</sup><sup>ϕ</sup>, (143)

, (144)

$$\Psi(\varkappa, t) = A \operatorname{cn} [B(\varkappa - \nu t), l] e^{i\phi}, \tag{146}$$

$$B = \left(\frac{b}{4d}\right)^{1/2},\tag{147}$$

$$
\rho = B^2 \left( 2dA^2 - a \right) - ak^2. \tag{148}
$$

Since the functions that we have introduced in these chapters comply with differential and algebraic equations similar to the ones for Jacobi's elliptic functions, we can give additional solutions in terms of these new functions, giving rise to new sets of soliton waves.
