Exact Traveling Wave Solutions for the System of Shallow Water Wave Equations and Modified Liouville Equation Using Extended Jacobian Elliptic Function Expansion Method

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the system of shallow water wave equations and modified Liouville equation which play an important role in mathematical physics.

Extended Jacobian Elliptic Function Expansion Method and Its Applications in Biology

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to Dynamical system in a new Double-Chain Model of DNA and a diffusive predator-prey system which play an important role in biology.

Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation

By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.

Jacobi Elliptic Function Expansion Method for the Nonlinear Vakhnenko Equation

By using Jacobi elliptic function expansion method, several kinds of travelling wave solutions of Nonlinear Vakhnenko equation are obtained in this paper. As a result, some new forms of traveling wave solutions of the equation are shown, and the numerical simulation with different parameters for the new forms solutions are given.

On a Generalized Extended F-Expansion Method

Making use of a new generalized ansatz, we present a new generalized extended F-expansion method for constructing the exact solutions of nonlinear partial differential equations in a unified way. Applying the generalized method with the aid of Maple, we consider the （2＋1）-dimentional breaking soliton equation. As a result, we successfully obtain some new and more general solutions including Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, and so on. As an illustrative sampler the properties of some soliton solutions for the breaking soliton equation are shown by some figures. Our method can also be applied to other partial differential equations.

General Solution of Two Generalized Form of Burgers Equation by Using the (<i>G</i>^'/<i>G</i>)-Expansion Method

In this work, the (G'/G)-expansion method is proposed for constructing more general exact solutions of two general form of Burgers type equation arising in fluid mechanics namely, Burgers-Korteweg-de Vries (Burgers-KdV) and Burger-Fisher equations. Our work is motivated by the fact that the (G'/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Jacobian Elliptic Function Method and Solitary Wave Solutions for Hybrid Lattice Equation

In this paper, we have successfully extended the Jacobian elliptic function expansion approach to nonlinear differential-difference equations. The Hybrid lattice equation is chosen to illustrate this approach. As a consequence, twelve families of Jacobian elliptic function solutions with different parameters of the Hybrid lattice equation are obtained. When the modulus m → 1 or O, doubly-periodic solutions degenerate to solitonic solutions and trigonometric function solutions, respectively.

New Extended Jacobi Elliptic Function Rational Expansion Method and Its Application

In this paper, an extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact Jacobi elliptic function solutions to nonlinear partial differential equations by means of making a more general transformation. For illustration, we apply the method to the （2＋1）-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions, which degenerate as soliton solutions when the modulus m approximates 1. The method can also be applied to other nonlinear partial differential equations.

A Generalized Tanh-Function Type Method and the(G'/G) -Expansion Method for Solving

In this paper a generalized tanh-function type method is proposed by using the idea of the transformed rational function method. We show that the (G'/G)?-expansion method is a special case of the generalized tanh-function type method, so the (G'/G)?-expansion method is considered as a special deformation application of the transformed rational function method. We demonstrate that all solutions obtained by the (G'/G)?-expansion method were found by the generalized tanh-function type method. As applications, we consider mKdV equation. Compared with the (G'/G) -expansion method, the generalized tanh-function type method gives new and more abundant solutions.

General Solution of Generalized (2+1)–Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the –Expansion Method

In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Applications of F-expansion method to the coupled KdV system

An extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics is presented, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by Jacobi elliptic functions for the coupled KdV equations are derived. In the limit cases, the solitary wave solutions and the other type of travelling wave solutions for the system are also obtained.

Exact solutions for the coupled Klein-Gordon-Schrǒdinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations （K-G-S equations） by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

New Jacobi Elliptic Function Solutions for the Generalized Nizhnik-Novikov-Veselov Equation

In this paper, a new generalized Jacobi elliptic function expansion method based upon four new Jacobi elliptic functions is described and abundant solutions of new Jacobi elliptic functions for the generalized Nizhnik-Novikov-Veselov equations are obtained. It is shown that the new method is much more powerful in finding new exact solutions to various kinds of nonlinear evolution equations in mathematical physics.

Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to Nonlinear SchrSdinger Equations with Variable Coefficient

In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we obtain explicit solutions of nonlinear Schrodinger equation with variable-coefficient. These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some are found for the first time. Six figures are given to illustrate some features of these solutions. The method can be applied to other nonlinear evolution equations in mathematical physics.

Some New Exact Solutions of Jacobian Elliptic Function of Petviashvili Equation

By using the modified mapping method, we find new exact solutions of the Petviashvili equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.

Symbolic Computation of Extended Jacobian Elliptic Function Algorithm for Nonlinear Differential-Different Equations

The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schrodinger equation. When the modulous m → 1or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.

Jacobi-Sobolev Orthogonal Polynomialsand Spectral Methods for Elliptic Boundary Value Problems

Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.

Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation

(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.

THE EXTENDED JACOBIAN ELLIPTIC FUNCTION EXPANSION METHOD AND ITS APPLICATIONS IN WEAKLY NONLINEAR WAVE EQUATIONS

The extended Jacobian elliptic function expansion method is introduced and applied to solve the coupled ZK equations and the coupled KP equations describing two weakly long nonlinear wave models in fluid system. Many types of doubly periodic traveling wave solutions are obtained. Under limiting conditions these solutions are reduced into solitary wave solutions.

Some New Exact Traveling Wave Solutions for the Generalized Benney-Luke (GBL) Equation with Any Order

In the paper, a auxiliary equation expansion method and its algorithm is proposed by studying a second order nonlinear ordinary differential equation with a four-degree term. The method is applied to the generalized Ben-ney-Luke (GBL) equation with any order. As a result, some new exact traveling wave solutions are obtained which singular solutions, triangular periodic wave solutions and jacobian elliptic function solutions .This algorithm can also be applied to other nonlinear wave equations in mathematical physics.