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Travelling wave solutions for the generalized Schrödinger equation

In this work, the generalized nonlinear Schrödinger equation is investigated. This equation is integrable and admits Lax pair. To obtain travelling wave solutions the extended tanh method is applied. This method is effective to obtain the exact solutions for different types of nonlinear partial differential equations. Graphs of obtained solutions are presented. The derived solutions are found to be important for the explanation of some practical physical problems.

The well-known soliton equation is the nonlinear Schrödinger (NLS) equation which is one of the most important and universal nonlinear models in modern science. Various NLS equations are widely used to describe different nonlinear physical problems, such as nonlinear optics [19][20], Bose-Einstein condensates [21], and others.
One of the simplest extensions of the NLS equation is the so-called generalized nonlinear Schrodinger (GNLS) equation [22][23], which is where q is a complex-valued function of the spatial coordinate z and the time t, the subscripts denote the partial derivatives with respect to the variables z, t. The GNLS equation (1) is a typical soliton equation with rich physical and mathematical applications where α denotes the amplification or absorption and γ relates to the group velocity [22][23]. In the case α = 0, γ = 0 in Eq. (1) we can get classical NLS equation. The Eq. (1) jointly with Maxwell-Bloch equations were studied in [24] where researchers found bound solitons and two types of breathers by the Darboux transformation. The aim of this paper is to construct some new exact solutions for the Eq. (1). We study the Eq. (1) by the extended tanh method that has been widely applied for a wide variety of nonlinear problems.

Travelling wave solutions
In this section we apply the extended tanh method to GNLS equation (1). By transformation the Eq. (1) can be converted to Separating real and imaginary parts in the Eq. (3) we obtain next system Substituting the wave transformation into system of Eqs. (4)-(5) we get From Eq. (7) we can derive Next, we study ordinary differential equation (8) u where prime denotes the derivation with respect to ξ and c is expressed by (9). To apply the extended tanh method we have to define a value M [7][8][9]. For this purpose, we balance the highest order derivative u , which has the exponent M + 2, with the nonlinear term u 3 , which has the exponent 3M in (10). It gives 3M = M + 2 that yields M = 1. Then, the extended tanh method let to apply the substitution where Y = tanh(µξ), ξ = z − ct, and µ is the wave number. Substituting (11) into (10) and collecting the coefficients of Y n , then we have a system of algebraic equations for µ, a 0 , a 1 , b 1 . By solving obtained system with the aid of Maple, we get the next results: Case 1: Case 3: Case 4: By substituting Eq. (11) with (6) into Eq. (2) we have general solutions as where ξ = z − ct.
Graphical presentation of the solutions (21)- (24) are shown in Figure 1- Figure 4 on the z − t plane with the parameters: a = 1; d = 1; α = 1; γ = 3. As we notice from 3D plots and density plots the solutions q 1 , q 3 give the bright solitons in contrast to q 4 that shows dark soliton. Moreover, periodic type solution q 2 is presented in Figure 2. Analyzing the graphs of obtained solutions we notice that the extended tanh method can yield various types of travelling wave solutions.

Conclusions
In this work, the extended tanh method have been applied to the generalized nonlinear Schrodinger equation. We obtained various type of the travelling wave solutions such as solitons, periodic solution. Moreover, the graphical presentation of the obtained solutions is shown in the figures. The obtained solutions can have an application to some practical physical problems.