Abstract and Applied Analysis

Volume 2017, Article ID 5137946, 8 pages

https://doi.org/10.1155/2017/5137946

## Three Different Methods for New Soliton Solutions of the Generalized NLS Equation

Al-Rafidain University College, Baghdad 00964, Iraq

Correspondence should be addressed to Anwar Ja’afar Mohamad Jawad; moc.oohay@1002dawaj_rawna

Received 23 June 2017; Revised 2 September 2017; Accepted 20 September 2017; Published 18 October 2017

Academic Editor: Jozef Banas

Copyright © 2017 Anwar Ja’afar Mohamad Jawad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Three different methods are applied to construct new types of solutions of nonlinear evolution equations. First, the Csch method is used to carry out the solutions; then the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions. The effectiveness of these methods is demonstrated by applications to the RKL model, the generalized derivative NLS equation. The solitary wave solutions and trigonometric function solutions are obtained. The obtained solutions are very useful in the nonlinear pulse propagation through optical fibers.

#### 1. Introduction

Partial differential equations describe various nonlinear phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology, and mathematical finance. Partial differential equations which arise in real-world physical problems are often too complicated to be solved exactly. It is of significant importance to solve nonlinear partial differential equations (NLPDEs) from both theoretical and practical points of view. The analysis of some physical phenomena is investigated by the exact solutions of nonlinear evolution equations (NLEEs) [1–9].

In this paper, the third-order generalized NLS equation is studied, which is proposed by Radhakrishnan, Kundu, and Lakshmanan (RKL) [10]. The normalized RKL model can be written as Equation (1) describes the propagation of femtosecond optical pulses, represents normalized complex slowly varying amplitude of the pulse envelope, and , and are real constants. Some solitary wave solutions and combined Jacobian elliptic function solution were constructed by different methods [3, 4].

The Csch method is used to carry out the solutions. Then, the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions of this equation.

#### 2. Traveling Wave Solution

Consider the nonlinear partial differential equation in the formwhere is a traveling wave solution of nonlinear partial differential equation (2). We use the transformations,where . This enables us to use the following changes:Using (4) to transfer the nonlinear partial differential equation (2) to nonlinear ordinary differential equation,The ordinary differential equation (5) is then integrated as long as all terms contain derivatives, where we neglect the integration constants.

#### 3. The Generalized NLS Equation (RKL)

In this section, the generalized third-order NLS equation (RKL) (1) is chosen to illustrate the effectiveness of three methods.

The solution of (1) may be supposed asSubstituting (6) into (1) and by defining the derivatives,then decomposing (1) into real and imaginary parts yields a pair of relations which represented nonlinear ordinary differential equations. The real part iswhile the imaginary part isIntegrating (9) once and setting the integration constant to zero, we obtainEquations (8) and (10) will be equivalent, provided thatfrom which we get the parametric constraintsmultiplying both sides of (10) by and integrating with respect to with zero constant, we getassume thatThen

#### 4. Methodology

In this section we will apply three different methods to solve (15). These methods are Csch method, Extended Tanh-Coth method, and the modified simple equation method (MSEM).

##### 4.1. Csch Function Method

The solution of many nonlinear equations can be expressed in the form [11]and their derivativewhere , , and are parameters to be determined and and are the wave number and the wave speed, respectively. We substitute (16)-(17) into the reduced equation (15); we getBalance the terms of the Csch functions to find We next collect all terms in (18) with the same power in and set their coefficients to zero to get a system of algebraic equations among the unknowns , , and and solve the subsequent systemSolving the system of equations in (20), we getthenthereforeFigure 1 represents the solitary wave in (23) for , , , , and then .