Exact Solutions for ( 2 + 1 )-dimensional Nonlinear Schrödinger Equation Based on Modified Extended tanh Method

In this paper, modified extended tanh method is used to construct more general exact solutions of a (2+1)-dimensional nonlinear Schrödinger equation. With the aid of Maple and Matlab software, we obtain exact explicit kink wave solutions, peakon wave solutions, periodic wave solutions and so on and their images.


Introduction
It is well known that Schrödinger equation is one of the most basic equations of quantum mechanics. It reflects the state of micro particle changing with time. As it is a powerful tool for solving non relativistic problems in atomic physics, it was widely used in the fields of atomic, molecular, solid state physics, nuclear physics, chemistry and so on. Recently, searching and constructing exact solutions of nonlinear partial differential 16 (NLPD) equation is very meaningful for it can describe the problems of mechanics, control process, ecological and economic system, chemical recycling system and epidemiological. In the past several decades, much efforts have been on this aspect and many useful methods have been proposed such as inverse scattering method, Jacobi elliptic function method, F-expansion method, Darboux transform, the sine-cosine method and the tanh method. The tanh method is widely used as it can find exact as well as approximate solutions in a systematic way. Subsequently, Fan has proposed an extended tanh method and obtained the travelling wave solutions that cannot be obtained by the tanh method. Based on this approach, we employed the modified extended tanh method to construct a series of exact travelling wave solutions of a (2+1)-dimensional nonlinear Schrödinger (NLS) equation as

The tanh Method
Here we review the modified extended tanh method.
The modified extended tanh method is developed by Malfliet in [10,11], and used in [12][13][14] among many others. Since all derivatives of a tanh can be represented by tanh itself, consider the general NLPDE say in two variables Now we consider its travelling ( ) ( ), or ct x + = ξ and the equation becomes an ordinary differential equation. We apply the following series expansion: where b is a parameter to be determined, The Riccati equation has the following general solutions:
Substituting all those situations into (9) respectively, we can get all solutions of the derivative nonlinear Schrödinger equation.

Image Simulation
In order to grasp these exact travelling solutions, we choose several exact solutions and use the Matlab software to simulate images. In the process of image simulation, the figures and value of parameters we selected show as follows:  In Figure 1, we take , In Figure 2, we take .  In Figure 3, we take .
In Figure   4, we take .  In Figure 5, we take .
In Figure 6,   In Figure 9, we take .
In Figure   10, we take .  In Figure 11, we take .

Conclusion
In this paper we studied the nonlinear Schrödinger equation by finding its exact travelling wave solutions through the modified extended tanh methods. With the aid of waveform graphs of the solutions, we can obtain the related properties of the equation. However, we can also use the method of dynamical systems to obtain the bounded solutions.