scholarly journals The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yun-Mei Zhao ◽  
Ying-Hui He ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Equation Method ◽  
Simplest Equation Method ◽  
Klein Gordon ◽  
Schrödinger's Equation ◽  
Wave Reduction

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

scholarly journals The Improved Riccati Equation Method and Exact Solutions to mZK Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Xiaofeng Li
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Rational Functions ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Equation Method ◽  
Hyperbolic Functions ◽  
Wave Solutions

We utilize the improved Riccati equation method to construct more general exact solutions to nonlinear equations. And we obtain the travelling wave solutions involving parameters, which are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. When the parameters are taken as special values, the method provides not only solitary wave solutions but also periodic waves solutions. The method appears to be easier and more convenient by means of a symbolic computation system. Of course, it is also effective to solve other nonlinear evolution equations in mathematical physics.

scholarly journals New exact traveling wave solutions for DS-I and DS-II equations

2012 ◽  
Vol 17 (3) ◽  
pp. 369-378 ◽  
Author(s):  
Ahmet Yildirim ◽  
Ameneh Samiei Paghaleh ◽  
Mohammad Mirzazadeh ◽  
Hossein Moosaei ◽  
Anjan Biswas
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Solution Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Equation Method ◽  
Wave Solutions ◽  
Simplest Equation Method

In this present work, the simplest equation method is used to construct exact solutions of the DS-I and DS-II equations. The simplest equation method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations. This method can be applied to nonintegrable equations as well as to integrable ones.

Method of Multiple Scales and Travelling Wave Solutions for (2+1)-Dimensional KdV Type Nonlinear Evolution Equations

2016 ◽  
Vol 71 (8) ◽  
pp. 703-713 ◽  
Author(s):  
Burcu Ayhan ◽  
M. Naci Özer ◽  
Ahmet Bekir
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Evolution ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Kdv Type Equations

AbstractIn this article, we applied the method of multiple scales for Korteweg–de Vries (KdV) type equations and we derived nonlinear Schrödinger (NLS) type equations. So we get a relation between KdV type equations and NLS type equations. In addition, exact solutions were found for KdV type equations. The$\left( {{{G'} \over G}} \right)$-expansion methods and the$\left( {{{G'} \over G},{\rm{ }}{1 \over G}} \right)$-expansion methods were proposed to establish new exact solutions for KdV type differential equations. We obtained periodic and hyperbolic function solutions for these equations. These methods are very effective for getting travelling wave solutions of nonlinear evolution equations (NEEs).

scholarly journals A modified generalized projective Riccati equation method

2016 ◽  
Vol 12 (6) ◽  
pp. 6318-6334
Author(s):  
Luwai Wazzan ◽  
Shafeek A Ghaleb
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Evolution Equations ◽  
Equation Method ◽  
New Exact Solutions ◽  
Special Cases ◽  
Projective Riccati Equation Method

A modification of the generalized projective Riccati equation method is proposed to treat some nonlinear evolution equations and obtain their exact solutions. Some known methods are obtained as special cases of the proposed method. In addition, the method is implemented to find new exact solutions for the well-known Dreinfelds-Sokolov-Wilson system of nonlinear partial differential equations.

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