An Extended Method for Constructing Travelling Wave Solutions to Nonlinear Partial Differential Equations

In this paper, an extended method is proposed for constructing new forms of exact travelling wave solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to the asymmetric Nizhnik-Novikov-Vesselov equation and the coupled Drinfel＇d-Sokolov-Wilson equation and successfully cover the previously known travelling wave solutions found by Chen＇s method .     

A Generalized Algebraic Method for Constructing a Series of Explicit Exact Solutions of a （1＋1）-Dimensional Dispersive Long Wave Equation

Making use of a new and more general ansatz, we present the generalized algebraic method to uniformly construct a series of new and general travelling wave solution for nonlinear partial differential equations. As an application of the method, we choose a (1 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by the method proposed by Fan [E. Fan, Comput. Phys. Commun. 153 (2003) 17] and find other new and more general solutions at the same time, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solutions, hyperbolic and soliton solutions, Jacobi and Weierstrass doubly periodic wave solutions.     

An Automated Jacobi Elliptic Function Method for Finding Periodic Wave Solutions to Nonlinear Evolution Equation

We describe the Jacobi elliptic function method for finding exact periodic wave solutions to nonlinear evolution equations.We present a Maple packaged automated Jacobi elliptic function method,which can entirely automatically output the exact periodic wave solutions.The effectiveness of the automated Jacobi elliptic function method is demonstrated using as examples the spplication to a variety of equations with physical interest.Not only are the previously known solutions recovered but in some cases new solutions and more general forms of solutions are obtained.     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimenslonal hyperbolic equations of conversation laww and the Burgers-KdV equation, a class of travellng wave solutions has been obtained by constructhag appropriate function transformations. The main idea of solving the equations is that nonlinear partial differential equations are changed into solving algebraic equations. This method has a wide-ranging practicability.     

Application of Modified G′/G-Expansion Method to Traveling Wave Solutions for Whitham-Broer-Kaup-Like Equations

A modified G′/G-expansion method is presented to derive traveling wave solutions for a class of nonlinear partial differential equations called Whitham -Broer- Kaup-Like equations. As a result, the hyperbolic function solutions, trigonometric function solutions, and rational solutions with parameters to the equations are obtained. When the parameters are taken as special values the solitary wave solutions can be obtained.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

New Exact Travelling Wave Solutions to Kundu Equation

Based on a first-order nonlinear ordinary differential equation with Six-degree nonlinear term, we first present a new auxiliary equation expansion method and its algorithm. Being concise and straightforward, the method is applied to the Kundu equation. As a result, some new exact travelling wave solutions are obtained, which include bright and dark solitary wave solutions, triangular periodic wave solutions, and singular solutions. This algorithm can also be applied to other nonlinear evolution equations in mathematical physics.     

Adomian Decomposition Method and Exact Solutions of the Perturbed KdV Equation

The Adomian decomposition method is used to solve the Cauchy problem of the perturbed KdV equation.Three types of exact solitary wave solutions are reobtained via the Adomian‘s approach by selcting the initial conditions appropriately.     

Solitary Wave and Non-traveling Wave Solutions to Two Nonlinear Evolution Equations

By applying the extended homogeneous balance method, we find some new explicit solutions to two nonlinear evolution equations, which include n-resonance plane solitary wave and non-traveling wave solutions.     

New Cnoidal and Solitary Wave Solutions of Coupled Higher-Order Nonlinear SchrSdinger System in Nonlinear Optics

The coupled higher-order nonlinear Schroedinger system is a major subject in nonlinear optics as one of the nonlinear partial differential equation which describes the propagation of optical pulses in optic fibers. By using coupled amplitude-phase formulation, a series of new exact cnoidal and solitary wave solutions with different parameters are obtained, which may have potential application in optical communication.