Abstract

We show abundant simply periodic solutions, trigonometric solutions, hyperbolic function solutions and Weierstrass elliptic solutions of the reduction of the cubic nonlinear Schrödinger equation by the complex method with Painlevé analysis, and some solutions appear to be new. At last, we give some computer simulations to illustrate our main results.

2010 Mathematics Subject Classification. Primary 30D35; Secondary 35C07.

1. Introduction

In this paper, we consider the cubic nonlinear Schrödinger equation (NLSE)

(1)

where is a complex envelop amplitude, is time, is distance, are velocity dispersions, is the coefficient of the cubic nonlinearity. Eq.(1) has been discussed in ^{1, 2, 3} by using the tanh method and the sine-cosine method, in ⁴ by using exp-function method, in ⁵ by several integration tools. In ⁶, Hong et al. researched on the general perturbed nonlinear Schrödinger equation by using the homotopy perturbation method. In ⁷, some analytical solutions were derived for the relevant case of of the time-dependent Schrödinger equation with the Riesz space-fractional derivative.

Recently, W. Yuan et al. ⁸ studied meromorphic solutions for some nonlinear differential equations based on the complex method and the Nevanlinna value distribution theory. Therefore, it is of interest to know whether the complex method can be applied to NLSEs. By using the traveling wave transformation , Eq.(1) reduces to

(2)

We study the meromorphic solutions of Eq.(2) on the complex plane, therefore, let .

2. Preliminaries

Consider the following algebraic ordinary differential equation

(3)

where is a polynomial in and its derivatives with constant coefficients.

If there are exactly distinct formal meromorphic Laurent series

(4)

satisfies Eq.(3), we say Eq.(3) satisfies condition ⁸.

If we only determine distinct principle parts , we say Eq.(3) satisfies weak condition.

Weierstrass elliptic function satisfies where the invariants and discriminant Furthermore, , , , any th derivatives of can be deduced by these identities, and has the Laurent series expansion

Defined ⁹ if is an elliptic function, or a rational function of , or a rational function of .

Lemma 2.1. ¹⁰ Let Suppose that equation (3) satisfies the condition, then all meromorphic solutions belong to the class . Furthermore, each elliptic solution with pole at can be written as

(5)

where are given by (4), and

Each rational function solution is of the form

(6)

with distinct poles of multiplicity .

Each simply periodic solution is a rational function of . has distinct poles of multiplicity , and is of the form

(7)

Definition 2.2. ¹⁰ Let be a meromorphic solution of a -th order algebraic differential equation . We call the involved term of which determining the multiplicity in as the dominant term. The dominant part of is consists of all dominant terms, and is denoted by . The multiplicity of pole of each term in is the same integer denoted by . The multiplicity of pole of each monomial in is denoted by .

Definition 2.3. ¹¹ For any meromorphic function , the derivative operator of dominant part with respect to is defined by

(8)

The roots of the following equation

(9)

is called the Fuchs index of .

The complex method ⁸ can be presented in the following five steps:

1) Substituting the transform

into a given PDE.

2) Substituting (4) into Eq.(3) to determine the (weak) condition holds.

3) By indeterminant relations (5-7) we find the elliptic, rational and simply periodic solutions of Eq.(3) with pole at , respectively.

4) By Lemma 2.1 we obtain all meromorphic solutions .

5) Substituting the inverse transform into these meromorphic solutions , then we get the exact traveling solutions of the original given PDE.

3. Main Results

Let be a meromorphic solution of Eq.(2), and suppose that has a movable pole at , then in a neighborhood of zero, the Laurent series of is of the form of . Substituting this Laurent series into Eq.(2), then

vanishing the coefficients of the lowest power , we have , and

From (2), we know that , therefore, for any meromorphic function ,

(10)

Hence, the Fuchs index equation of Eq.(2) reads

It is easily to know that . By the Painlevé analysis ¹¹ we know that there is an arbitrary coefficient for some in the Laurent series . Therefore, Eq.(2) satisfies the weak condition, and then we will build meromorphic solutions for Eq.(2) by Lemma 2.1.

By (6), we infer the indeterminant rational solutions of Eq.(2):

Substituting into Eq.(2), combining similar terms, then vanishing all coefficients to zero, we build the following rational solutions:

(12)

where , , and is an arbitrary number.

Noting that , by Lemma 2.1, we infer that the indeterminant of elliptic solution of Eq.(2) with pole at zero:

Substituting into Eq.(2), combining similar terms, then vanishing all coefficients to zero, then we build the following system of algebraic equations:

(13)

Solving (13), we have ,

Therefore, we obtain the following elliptic function solutions of Eq.(2) with pole at :

(14)

where

(15)

and is an arbitrary number.

Assuming that , we have

then , or . Furthermore, we have

(16)

then the elliptic solutions (14) can be degenerated to the following trigonometric function solutions:

(17)

(18)

where .

By (7), we infer the indeterminant simply periodic solutions of Eq.(2):

Put into Eq.(2), vanishing all coefficients to zero, we build following simply periodic solutions:

(19)

If , then , (19) will be reduce to a constant.

Furthermore, we investigate the special cases of (19) with poles at and . By and (19), we can obtain the following hyperbolic function solutions:

(20)

(21)

(22)

(23)

where , and can be replaced by , is an arbitrary number.

It follows easily that the solitons (22) (23) can be reduced to real functions with initial condition and as follows:

(24)

(25)

Noting that Eq.(2), if is a solution, then is also.

Remark 1. are the traveling wave solutions satisfy Eq.(1). The solutions (20)-(25) are also appear in ² and ³.p690, and solutions (12) (14) (16) (17) (18) (19) are appear to be new comparing to ^{1, 2} and other open literatures.

4. Computer Simulations

In this section, we give some computer simulations to illustrate the main results. If are fixed, the amplitude and the width of the solitons (21) (25) can be manipulated using the dispersion coefficients . We fix in the solitons (21) (25), see Figure 1 and Figure 2 for the values of for bright solitons and for dark solitons. If is big the amplitude are small, and if is small the solitons are thin, while increasing the dispersion coefficient , their widths are also increasing if the solitons spread. The figure of Weierstrass function solution (14) be shown as Figure 3 for the values of and .

5. Conclusions

This work is a successive application of the complex method for constructing exact solutions on the cubic nonlinear Schrödinger equation, elliptic function solutions, simply periodic solutions, trigonometric function solutions and hyperbolic function solutions were investigated. The results show that the complex method is a powerful and systematic tool for constructing meromorphic solutions for some certain complex ordinary differential equations, especially for solitons and periodic solutions.

Acknowledgments

This work was partial supported by the visiting scholar program of Shiing-Shen Chern Institute of Mathematics at Nankai University. The author is also thankful to the referees for their invaluable comments and suggestions, which put the article in its present shape.

References