Numerical solution of the generalized Zakharov equation by homotopy analysis method

In this paper, an analytic technique, namely the homotopy analysis method (HAM) is applied to obtain approximations to the analytic solution of the generalized Zakharov equation. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of the solution series.     

A series of abundant exact travelling wave solutions for a modified generalized Vakhnenko equation using auxiliary equation method

In this paper, a series of abundant exact travelling wave solutions is established for a modified generalized Vakhnenko equation by using auxiliary equation method. These solutions can be expressed by Jacobi elliptic function. When Jacobi elliptic functions modulus m→1 or 0, the travelling wave solutions degenerate to four types of solutions, namely, the soliton solutions, the hyperbolic function solutions, the trigonometric function solutions, constant solutions.     

应用拓展双曲函数方法求KP方程的新精确解

本文引入行波解,并应用拓展双曲函数方法,求得（2＋1）维Kadomtsev-Petviashvili（KP）方程的精确解.通过应用拓展双曲函数方法,可以得到关于方程的一类有理函数形式的孤立波,行波以及三角函数周期波的精确解,并且此方法适用于求解一大类非线性偏微分进化方程.     

The exponential function rational expansion method and exact solutions to nonlinear lattice equations system

We propose an exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential-difference equations system. By this method, we obtain some exact traveling wave solutions to the relativistic Toda lattice equations system and discuss the significance of these solutions. Finally, we give an open problem.     

A note on new exact solutions for the Kawahara equation using Exp-function method

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, New Exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong.     

On the existence and uniqueness of smooth solution for a generalized Zakharov equation

The paper deals with the existence and uniqueness of smooth solution for a generalized Zakharov equation. We establish local in time existence and uniqueness in the case of dimension d=2,3. Moreover, by using the conservation laws and Brezis-Gallouet inequality, the solution can be extended globally in time in two dimensional case for small initial data. Besides, we also prove global existence of smooth solution in one spatial dimension without any small assumption for initial data.     

A note on rational solutions to a Hirota-Satsuma-like equation

With the generalized bilinear operators based on a prime number $p=3$, a Hirota-Satsuma-like equation is proposed. Rational solutions are generated and graphically described by using symbolic computation software Maple.     

Three types of exact solutions to Wick-type generalized stochastic Korteweg-de Vries equation

A Wick-type generalized stochastic Korteweg-de Vries equation is researched. By means of Hermite transformation, white noise theory and Riccati equation mapping method, three types of exact solutions to the generalized stochastic Korteweg-de Vries equation, which include the functional solutions of hyperbolic-exponential type, trigonometric-exponential type and exponential type, are derived.     

A series of exact solutions for (2 + 1)-dimensional Wick-type stochastic generalized Broer-Kaup system via a modified variable-coefficient projective Riccati equation mapping method

A modified variable-coefficient projective Riccati equation mapping method is applied to (2 + 1)-dimensional Wick-type stochastic generalized Broer-Kaup system. With the help of Hermit transformation, we obtain a series of new exact stochastic solutions to the stochastic Broer-Kaup system in the white noise environment.     

The extended hyperbolic functions method and new exact solutions to the Zakharov equations

The multiple exact solutions for the nonlinear evolution equations describing the interaction of laser–plasma are developed. The extended hyperbolic function method are employed to reveal these new solutions. The solutions include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the solitary wave solutions of a compound of the bell-type and the kink-type for n and E, the singular traveling wave solutions, periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. In addition to re-deriving all known solutions in a systematic way, several new and more general solutions can be obtained by using our method.     

A new method for a generalized Hirota-Satsuma coupled KdV equation

In this paper, differential transform method (DTM), which is one of the approximate methods is implemented for solving the nonlinear Hirota-Satsuma coupled KdV partial differential equation. A variety of initial value system is considered, and the convergence of the method as applied to the Hirota-Satsuma coupled KdV equation is illustrated numerically. The obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient. Numerical examples are illustrated the pertinent features of the proposed algorithm.     

New exact solutions for a generalized variable-coefficient KdV equation

New exact solutions for a generalized variable-coefficient KdV equation were obtained using the generalized expansion method [R. Sabry, M.A. Zahran, E.G. Fan, Phys. Lett. A 326 (2004) 93]. The obtained solutions include solitary wave solutions besides Jacobi and Weierstrass doubly periodic wave solutions.     

Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation

In this paper, by using the integral bifurcation method, we study a generalized KdV equation which was first derived by Fokas from physical considerations via a methodology of Fuchssteiner. All kinds of soliton-like or kink-like wave solutions and periodic wave solutions with loop or without loop are obtained. Smooth compacton-like periodic wave solution and non-smooth periodic cusp wave solution are also obtained. Their dynamic properties are investigated and their profiles are given by Mathematical software.     

New exact solutions to the generalized Burgers-Huxley equation

Based on He’s Exp-function method, a series of new exact solutions of the generalized Burger-Huxley equation have been obtained. It is shown that the Exp-function method is straightforward and concise, and its applications are promising.     

Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1 + 1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solution with arbitrary functions for the PKPp.     

Symmetry reductions and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1+1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solutions with arbitrary functions for the PKPp.     

Group classifications, optimal systems and exact solutions to the generalized Thomas equations

In this paper, the complete group classifications are performed on the types of Thomas equations (TEs), which arise in the study of chemical exchange progress, etc., all of the vector fields of the equations are presented. Then, the optimal system of the general Thomas equation is given, and all of the symmetry reductions and exact solutions generated from the optimal system are investigated. Furthermore, the exact analytic solutions to the Thomas equations are obtained by the generalized power series method.     

New exact solutions for mCH and mDP equations by auxiliary equation method

With the aid of symbolic computation, auxiliary equation method is introduced to investigate modified forms of Camassa-Holm and Degasperis-Procesi equations. A series of new exact traveling wave solutions, including smooth solitary wave solution, peakons, singular solution, periodic wave solution, Jacobi elliptic solution, are obtained in general form. These new exact solutions will enrich previous results and help us further understand the physical structures of these two nonlinear equations.     

广义组合KdV-mKdV方程的显式精确解

Abstract. With the aid of Mathematica and Wu-elimination method,via using a new generalizedansatz and well-known Riccati equation,thirty-two families of explicit and exact solutions forthe generalized combined KdV and mKdV equation are obtained,which contain new solitarywave solutions and periodic wave solutions. This approach can also be applied to other nonlinearevolution equations.