Evolution Equations In Mathematical Physics Research Articles

Cross-kink multi-soliton solutions for the (3+1)-D Jimbo-Miwa equation

In this paper, by using bilinear form and extended three-wave type of ans¨atz approach, we obtain new cross-kink multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breather-type of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breathertype of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics.

Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

We use extended mapping method and auxiliary equation method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solution for the Boussinesq system and the coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

THE EXTENDED RICCATI EQUATION METHOD FOR TRAVELLING WAVE SOLUTIONS OF ZK EQUATION

In this article, the extended Riccati equation method is applied to seeking more general exact travelling wave solutions of the ZK equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values, the solitary wave solutions are obtained from the hyperbolic function solutions. Similarly, the periodic wave solutions are also obtained from the trigonometric function solutions. The approach developed in this paper is effective and it may also be used for solving many other nonlinear evolution equations in mathematical physics.

The Two‐Variable (G′/G, 1/G)‐Expansion Method for Solving the Nonlinear KdV‐mKdV Equation

We apply the two‐variable (G′/G, 1/G)‐expansion method to construct new exact traveling wave solutions with parameters of the nonlinear (1 + 1)‐dimensional KdV‐mKdV equation. This method can be thought of as the generalization of the well‐known (G′/G)‐expansion method given recently by M. Wang et al. When the parameters are replaced by special values, the well‐known solitary wave solutions of this equation are rediscovered from the traveling waves. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

The Improved Riccati Equation Method and Exact Solutions to mZK Equation

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.

The modified alternative -expansion method for finding the exact solutions of nonlinear PDEs in mathematical physics

In this article, we construct new exact traveling wave solutions of some nonlinear evolution equations in mathematical physics with the aid of the modified alternative –expansion method. The nonlinear equations under investigation include, the (1+1)-dimensional Drinfeld-Sokolov-Wilson (DSW) equation, the travelling regularised long wave (TRLW) equation, the Caudrey-Dodd-Gibbon (CDG) equation and the (2+1)-dimensional Nizhnik-Novikov-Vesselov (NNV) equation. In the alternative –expansion method, satisfies the Riccati equation where and are arbitrary constants. The obtained solutions are expressed in terms of hyperbolic function, trigonometric function and rational functions. Key words: The -expansion method, the Riccati equation, the traveling wave solutions, the Drinfeld-Sokolov-Wilson equation, the travelling regularised long wave equation, the Caudrey-Dodd-Gibbon equation, the Nizhnik-Novikov-Vesselov equation.

Exact Multi-soliton Solution for the (3+1)-D Jimbo-Miwa Equation

In this paper, by using bilinear form and extended three-wave type of ans$\ddot{a}$tz approach, we obtain new multi-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation, including the periodic breather-type of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breather-type of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics.

The extended auxiliary equation method for the KdV equation with variable coefficients

This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.

Application of the $(G^{\prime}$ / $G)$ -expansion method for the Burgers, Burgers–Huxley and modified Burgers–KdV equations

In this work, we present travelling wave solutions for the Burgers, Burgers–Huxley and modified Burgers–KdV equations. The (G′/G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

New exact solutions of coupled (2+1)-dimensional nonlinear systems of Schrödinger equations

The Exp-function method is applied to construct a new type of solution of the coupled (2+1)-dimensional nonlinear system of Schrödinger equations. It is shown that the method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. doi:10.1017/S1446181111000563

The Modified (G'/G)-Expansion Method for Nonlinear Evolution Equations

A modified (Gʹ/G)-expansion method is proposed to construct exact solutions of nonlinear evolution equations. To illustrate the validity and advantages of the method, the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is considered and more general travelling wave solutions are obtained. Some of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions, and rational solutions contain an explicit linear function of the variables in the considered equation. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Analytical Method in Solving Flow of Viscoelastic Fluid in a Porous Converging Channel

An analytical method, called homotopy perturbation method (HPM), is used to compute an approximation to the solution of the nonlinear differential equation governing the problem of two‐dimensional and steady flow of a second‐grade fluid in a converging channel. The table and figures are presented for influencing various parameters on the velocity field. The results compare well with those obtained by the numerical method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

Travelling Wave Solutions for the Burgers Equation and the Korteweg-de Vries Equation with Variable Coefficients Using the Generalized (G´/G)-Expansion Method

In this article, a generalized (G´/G)-expansion method is used to find exact travelling wave solutions of the Burgers equation and the Korteweg-de Vries (KdV) equation with variable coefficients. As a result, hyperbolic, trigonometric, and rational function solutions with parameters are obtained. When these parameters are taking special values, the solitary wave solutions are derived from the hyperbolic function solution. It is shown that the proposed method is direct, effective, and can be applied to many other nonlinear evolution equations in mathematical physics.

New Soliton and Periodic Solutions for Two Nonlinear Physical Models

In this paper, the exp-function method is applied by using symbolic computation to construct a variety of new generalized solitonary and periodic solutions for the Burgers-Kadomtsev-Petviashvili and Vakhnenko equations with distinct physical structures. The results reveal that the exp-function method is very effective and powerful for solving nonlinear evolution equations in mathematical physics.

Exact three-wave solutions for the [formula omitted]-dimensional Jimbo–Miwa equation

Exact three-wave solutions including periodic cross-kink wave solutions, doubly periodic solitary wave solutions and breather type of two-solitary wave solutions for the (3+1)-dimensional Jimbo–Miwa equation are obtained using the generalized three-wave method. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving high-dimensional nonlinear evolution equations in mathematical physics.

An Exp-function method for new [formula omitted]-soliton solutions with arbitrary functions of a [formula omitted]-dimensional vcBK system

In this paper, the Exp-function method is generalized to construct N -soliton solutions of the (2+1)-dimensional variable-coefficient Broer–Kaup (vcBK) system. As a result, new and more general 1-soliton, 2-soliton and 3-soliton solutions with arbitrary functions are obtained, from which the uniform formulae of N -soliton solutions are derived. Based on one of the obtained 2-soliton solution, the fission and fusion phenomena between the peakons and solitary waves are investigated. It is shown that the Exp-function method may provide us with a straightforward, effective and alternative mathematical tool for generating N -soliton solutions of nonlinear evolution equations in mathematical physics.

A series of new exact solutions of nonlinear evolution equations

AbstractWith the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

NEW EXACT SOLUTIONS OF COUPLED (2+1)-DIMENSIONAL NONLINEAR SYSTEMS OF SCHRÖDINGER EQUATIONS

AbstractThe Exp-function method is applied to construct a new type of solution of the coupled (2+1)-dimensional nonlinear system of Schrödinger equations. It is shown that the method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solutions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.

Exact Solutions for the Modified KdV and the Generalized KdV Equations via Exp-Function Method

An application of the Exp-function method (EFM) to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for the modified KdV equation and the generalized KdV equation. The EFM was used to construct peri- odic wave and solitary wave solutions of nonlinear evolution equations (NLEEs). This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics and applied mathematics.