Exact solutions to nonlinear Schrödinger equation with variable coefficients

doi:10.1016/j.amc.2010.12.072

Applied Mathematics and Computation

Volume 217, Issue 12, 15 February 2011, Pages 5866-5869

https://doi.org/10.1016/j.amc.2010.12.072 Get rights and content

Abstract

According to Ma-Fuchsseiter’s idea, a trial equation method was proposed to find the exact envelop traveling wave solutions to some nonlinear differential equations with variable coefficients. As an application, combining with the complete discrimination system for polynomial, some exact envelop traveling wave solutions to Schrödinger equation with variable coefficients were obtained. At the same time, the physical meanings of the obtained solutions are discussed, and the problem needed to further study is pointed out.

Introduction

The differential equations with variable coefficients are more suitable to practice models. Varied method have been proposed to solve these equations, and many exact solutions were obtained (see, for example, Refs. [1], [2], [3], [4], [5], [6], [7]and the references therein). In 1996, Ma and Fuchssteiner proposed a powerful approach for finding exact solutions to nonlinear differential equations [8]. Their key idea is to expand solutions of given differential equations as functions of solutions of solvable differential equations, in particular, polynomial and rational functions. This idea is so important that many types of nonlinear equations can be solved by it. A more systematical theory on decompositions and transformations is presented very recently in Refs. [9], [10]. Ma and his coauthors’ theory unifies many existing approaches to exact solutions such as the tanh-function methods, the homogeneous balance method, the exp-function method and the Jacobi elliptic function method. On the other hand, Liu’s trial equation method [11], [12], [13], [14] is an efficient method to solve real nonlinear differential equations, many exact single traveling wave solutions have been obtained for a lot of nonlinear differential equations including variable coefficients equations. It is meaningful to use Ma-Fuchsseiter’s idea and generalize Liu’s method to find other type solution such as the envelop traveling wave solutions to some equations with variable coefficients. In the present paper, according to Ma-Fuchsseiter’s idea and combining with the complete discrimination system for polynomial method [15], we propose a trial equation method to obtain some exact envelope traveling wave solutions to nonlinear Schrödinger equation which is a complex equation with variable coefficients [4]. At the same time, the problem needed to further study is pointed out.

Section snippets

Trial equation method

According to Ma-Fuchsseiter’s idea and Liu’s trial equation method to real equation, we propose a trial equation method which can be suitable to both real equations and complex equations with variable coefficients.

Step 1.
We consider the following nonlinear differential equation with variable coefficients: $E (t, x, u, u_{t}, u_{x}, u_{tt}, u_{xt}, u_{xx}, \dots) = 0 .$ Take traveling wave transformation for real equation as follows: $u (x, t) = u (ξ), ξ = k (t) x + ω (t),$ or envelope traveling wave transformation for complex equation as follows: $u (x, t) =$

Application

We consider the Schrödinger equation with variable coefficients ([4]) ${iu}_{t} + f (t) u_{xx} + g (t) u | u |^{2} = 0 .$ Substituting envelope traveling wave transformation (3) into Eq. (9) and separating the real and imaginary parts yield two equations as follows: $[k^{'} (t) x + ω^{'} (t) + 2 f (t) s (t) k (t)] u^{'} = 0,$ $f (t) k^{2} (t) u^{″} - {s^{'} (t) x + r^{'} (t) + f (t) s^{2} (t)} u + g (t) u^{3} = 0 .$ Eliminating u′ in Eq. (10) and substituting trial Eq. (7) into Eq. (11), according to the balance principle we get m = 4. Corresponding ordinary differential equations system is given

Conclusions

We proposed a trial equation method for solving the exact solutions to Schrödinger equation with variable coefficients, some exact solutions were obtained. There are other differential equations with variable coefficients such as Schrödinger equation with more general variable coefficients ${iu}_{t} + f (x, t) u_{xx} + g (x, t) u | u |^{2} = 0;$ Zakharrov–Kuznetsov equation with variable coefficients $u_{t} + f (x, y, t) {uu}_{x} + g (x, y, t) u_{xxx} + h (x, y, t) u_{xxy} = 0,$ and KdV-type equations with variable coefficients, can be dealt with by the

Acknowledgments

The author thanks the referees for their valuable suggestions.

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Exact solutions to nonlinear Schrödinger equation with variable coefficients

Abstract

Introduction

Section snippets

Trial equation method

Application

Conclusions

Acknowledgments

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Phys. Lett. A.

Phys. Lett. A.

Int. J. Non-Linear Mech.

Phys. Lett. A

Chaos, Soliton Fractals