Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

doi:10.1016/j.amc.2009.01.002

Applied Mathematics and Computation

Volume 210, Issue 2, 15 April 2009, Pages 422-435

https://doi.org/10.1016/j.amc.2009.01.002 Get rights and content

Abstract

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.

Introduction

The nonlinear partial differential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in physics, chemistry, biology, fluid dynamics, plasma, optical fibers and other areas of engineering. Many efforts have been made to study NPDEs. One of the most exciting advances of nonlinear science and theoretical physics has been a development of methods that look for exact solutions for nonlinear evolution equations. The availability of symbolic computations such as Mathematica or Maple, has popularized direct seeking for exact solutions of nonlinear equations.

Therefore, exact solution methods of nonlinear evolution equations have become more and more important resulting in methods like Variation Iteration Method [1], [2], Homotopy Perturbation Method [3], [4], [5], [6], [7], [8], Exp-Function method [9], [10], [11], [12], [13], [14], [15], [36], [37], the sine–cosine method [16], [17], [18], the homogeneous balance method [19], tanh–sech method [20], [21], [22], [23], [24] and Extended tanh–coth method [25], [26], [27], [28], [29]. Most of exact solutions have been obtained by these methods, including the solitary wave solutions, shock wave solutions, periodic wave solutions, and the like.

In this paper, we propose Extended tanh–coth, sine–cosine and Exp-Function methods to obtain an exact single-soliton and travelling wave solutions of the generalized nonlinear Schrödinger (GNLS) equation with a source. In order to illustrate the effectiveness and convenience of these methods, we consider the GNLS equation in the form [30], [31], ${iu}_{t} + {au}_{xx} + bu | u |^{2} + {icu}_{xxx} + id (u | u |^{2})_{x} = {ke}^{i [χ (ξ) - wt]},$ where ξ = α(x − vt) is a real function and a, b, c, d, k, α, v and w are all real.

The GNLS equation (1.1) plays an important role in many nonlinear sciences. It arises as an asymptotic limit for a slowly varying dispersive wave envelope in a nonlinear medium. For example, its significant application in optical soliton communication plasma physics has been proved.

Furthermore, the GNLS equation enjoys many remarkable properties (e.g., bright and dark soliton solutions, Lax pair, Liouvile integrability, inverse scattering transformation, conservation laws, Backlund transformation, etc.).

The rest of this paper is as follows: in Sections 2 Tanh method and Extended tanh method, 3 Sine–cosine method, 4 Summary of Exp-Function method, we provide, in a simple way, the mathematical framework of Extended tanh–coth, sine–cosine and Exp-Function methods, respectively. In Section 5, in order to illustrate the application of these methods, generalized nonlinear Schrödinger equation with a source is investigated, and several exact solutions, including soliton like solutions and trigonometric function solutions, are obtained. This paper is concluded in the last section.

Section snippets

Tanh method and Extended tanh method

We now describe the tanh method for a given partial differential equation. This method was defined by Malfliet [20] and Fan and Hon [26]. Wazwaz summarized the main steps of using this method as follows [24]:

I.
Wazwaz first considered a general form of nonlinear equation: $N (u, u_{t}, u_{x}, u_{xx}, \dots) = 0,$
II.
To find the travelling wave solution of Eq. (2.1), he introduced the wave variable: $ξ = k (x + λ t),$ so that $u (x, t) = U (ξ),$ Therefore, Eq. (2.1) constructs ODE of form: $N (U, k λ U^{'}, {kU}^{'}, k^{2} U^{″}, \dots) = 0 .$
III.
If all terms of the resulting

Sine–cosine method

Wazwaz has summarized the main steps of using sine–cosine method, as listed below:

I.
Introducing the wave variables ξ = x − ct into the PDE, the following function is obtained: $ϕ (u, u_{t}, u_{x}, u_{tt}, u_{xx}, u_{xt}, u_{xxx} \dots) = 0,$ where u(x, t) is travelling wave solution. This allows the following changes: $u (x, t) = U (ξ),$ $\begin{matrix} \frac{\partial}{\partial t} = - c \frac{d}{d ξ}, \frac{\partial^{2}}{\partial t^{2}} = c^{2} \frac{d^{2}}{d ξ^{2}}, \\ \frac{\partial}{\partial x} = \frac{d}{d ξ}, \frac{\partial^{2}}{\partial x^{2}} = \frac{d^{2}}{d ξ^{2}}, \dots \end{matrix}$ and so for the other derivates. Using Eqs. (3.3), (3.1), the nonlinear PDE (3.1) is changed to a nonlinear ODE: $N (U, - {cU}^{'}, U^{'}, c^{2} U^{″}, U^{″}, - {cU}^{'}, U^{‴} \dots) = 0 .$
II.
If all terms of the

Summary of Exp-Function method

The Exp-Function method was first proposed by Wu and He in 2006 [10] and systematically studied in [32]. Furthermore, through the investigation of more than 280 references, He [36] presented an excellent study on the concepts of the recently developed asymptotic methods including Exp-Function. In addition, this method was successfully applied to KdV equation with variable coefficients [33], high-dimensional nonlinear evolution equation [14], Burgers and combine KdV–mKdV (Extended KdV) equations

Using Extended tanh method

To study the exact travelling wave solutions of the GNLS, Eq. (1.1), we consider a plane wave transformation in this form: $u (x, t) = ψ (ξ) e^{i [χ (ξ) - wt]},$ where ψ(ξ) is a real function. For convenience, let χ = βξ + x₀ where β and x₀ are real constants and ξ = α(x − vt). Then, by replacing Eq. (5.1) and its appropriate derivatives in Eq. (1.1) and separating the real and imaginary parts of the result, we obtain the two following ordinary differential equations: $c α^{3} ψ^{‴} + (- α v + 2 a β α^{2} - 3 c α^{3} β^{2}) ψ^{'} + 3 d α ψ^{2} ψ^{'} = 0,$ $(a α^{2} - 3 c ψ^{3} β) ψ^{″} + (α$

Conclusion

In this study, Extended tanh-function, sine–cosine and Exp-Function methods were applied in order to find the exact solutions of the generalized nonlinear Schrödinger (GNLS) equation with a source. The first two procedures can be considered as simple methods for solving some of nonlinear partial differential equations without resort to any symbolic computation and the obtained results are very concise. On the other hand, it can be concluded that, in comparison with the other methods, the