Exact Travelling Wave Solutions of Two Nonlinear Schrödinger Equations by Using Two Methods

The special kind of (G’/G)-expansion method and the new mapping method are easy and significant mathematical methods. In this paper, exact travelling wave solutions of the higher order dispersive Cubic-quintic nonlinear Schrodinger equation and the generalized nonlinear Schrodinger equation are studied by using the two methods. Finally, the solitary wave solutions, singular soliton solutions, bright and dark soliton solutions and periodic solutions of the two nonlinear Schrodinger equations are obtained. The results show that this method is effective for solving exact solutions of nonlinear partial differential equations.


Introduction
The nonlinear PDE is an important model for describing the problems of Nonlinear phenomenon, such as hydrodynamics, plasma physics, chemical dynamics, photobiology, solid physics, Marine and atmospheric phenomena, and so on. It can be seen from these fields that the travelling wave solutions of nonlinear evolution equations play an important role in the study. In order to find the exact solutions of nonlinear partial differential Equations (PDEs), pioneers presented the following these methods, such as the first integral method [1], Jacobi elliptic function expansion method [2], F expansion method [3], exp-function method [4], the Kudryashov method [5], the improved ( ) G G ′ -expansion method [6], In 2014, Kudryashov [16] substantiated that the ( ) introduced this method, see [19] [20] [21], and gave the specific solving process for nonlinear PDE. For the higher order dispersive Cubic-quintic NLSE, In 2017, Zayed and Nowehy [22] incorporated the solution Ansatz method with the Jacobi elliptic equation method to obtain several integrations denoted Jacobi elliptic function of the equation. In 2017, Arshad, sedawy and Lu [23] used an improved direct algebraic extension method to present bright and dark wave solutions and soliton wave solutions of higher order dispersive Cubic-quintic NLSEs. In addition, there is an amount of paper [24] [25] [26] where the various types of the equation are studied. For the GNLSE, In 2010, Geng and Li by using the dynamic system method and bifurcation theory, studies the travelling wave solution of the GNLSE and high order dispersion NLSE. the solitary wave solutions, kink and reverse kink wave solutions and periodic wave solutions are obtained. In 2007, Huang, Li and Zhang [27] through the study a class of nonlinear term six times of first order nonlinear ODE and applies it to the GNLSE. New accurate traveling wave solutions, such as light and dark isolated wave solutions, triangular periodic wave solutions and singular solutions are obtained. In addition, this GNLSE was studied, see [28] [29] [30].
The rest of the article is organized as follows: Section 2, we mainly describe the basic idea of the special kind of ( ) G G ′ -expansion method and the new mapping method briefly. In Section 3 and 4, we use these two methods to solve two NLSEs in detail. Some conclusions are drawn in Section 4.

Introduction of Two Methods
Method 1: The special kind of ( ) where P is a polynomial in its arguments.
In order to transform the Equation (3) into an ODE, we suppose that where c is a constant, then Step 1: According to above supposing, the Equation (3) has the following nonlinear ODE form: where the subscript denotes the derivation with respect to ξ .
Step 2: Suppose that the Equation (6) has non-integer balance number N. the solution of the Equation (6) can be written in the following special form, see [31] [32] [33]: where ( ) G ξ satisfies the linear ODE: Step 3: Firstly, determining the balance number N by balancing the high order derivative and the highest power of the nonlinear term in Equation (6).
Step 4: Then, substituting the Equations (7) and (8) into the Equation (6), and make the coefficients of ( ) ( ) all zero, and get a set of algebraic equations, which can be solved by Maple software to find , , ,c λ µ Ω .
Step 5: Finally by solving Equation (8) for exact solutions are obtained.

Method 2:
The new mapping method.
Step 1: We suppose that the Equation (6) has the formal solution: where F is an appropriate variable transformation, and ( ) ϕ ξ satisfies the following equation: where , , , δ α β γ are arbitrary constant to be determined.

Method
In this section, we apply the special kind of ( ) G G ′ -expansion method to solve the two higher order NLSE.

Application of the New Mapping Method
In this section, we apply the new mapping method to solve the two higher order NLSE.