求Nizhnik-Novikov-Veselov方程新精确解

引入改进的F-广义方法,并将其应用于（2＋1）维Nizhnik-Novikov-Veselov（NNV）方程.在符号计算软件的帮助下,可以得到NNV方程的许多新解.该方法用于获取包括雅可比椭圆函数解的一系列解,在数学物理中可应用于其他的非线性偏微分方程.     

Exact soliton solutions for the general fifth Korteweg-de Vries equation

With the aid of computer symbolic computation system such as Maple, the extended hyperbolic function method and the Hirota’s bilinear formalism combined with the simplified Hereman form are applied to determine the soliton solutions for the general fifth-order KdV equation. Several new soliton solutions can be obtained if we taking parameters properly in these solutions. The employed methods are straightforward and concise, and they can also be applied to other nonlinear evolution equations in mathematical physics. The article is published in the original.     

New exact solutions to some variable coefficients problems

With the aid of computer symbolic computation system such as Maple, an extended tanh method is applied to determine the exact solutions for some nonlinear problems with variable coefficients. Several new soliton solutions and periodic solutions can be obtained if we taking paraments properly in these solutions. The employed methods are straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations

In this paper, by using the improved Riccati equations method, we obtain several types of exact traveling wave solutions of breaking soliton equations and Whitham-Broer-Kaup equations. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method employed here can also be applied to solve more nonlinear evolution equations.     

A variety of exact travelling wave solutions for the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation

In this paper the (2 + 1)-dimensional Boiti-Leon-Pempinelli (BLP) equation will be studied. The tanh-coth method will be used to obtain exact travelling wave solutions for this equation. The Exp-function method will also be applied to the BLP equation to derive a new variety of travelling wave solutions with distinct physical structures.     

Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity

Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection-diffusion-reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection-diffusion-reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena.     

Exact solutions for nonlinear partial differential equations via Exp‐function method

In this article, the Exp‐function method is applied to nonlinear Burgers equation and special fifth‐order partial differential equation. Using this method, we obtain exact solutions for these equations. The method is straightforward and concise, and its applications are promising. This method can be used as an alternative to obtain analytical and approximate solutions of different types of nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Discrete exact solutions of modified Volterra and Volterra lattice equations via the new discrete sine-Gordon expansion algorithm

Recently, we have presented a sine-Gordon expansion method to construct new exact solutions of a wide of continuous nonlinear evolution equations. In this paper we further develop the method to be the discrete sine-Gordon expansion method in nonlinear differential-difference equations, in particular, discrete soliton equations. We choose the modified Volterra lattice and Volterra lattice equation to illustrate the new method such as many types of new exact solutions are obtained. Moreover some figures display the profiles of the obtained solutions. Our method can be also applied to other discrete soliton equations.     

Single and multi-solitary wave solutions to a class of nonlinear evolution equations

In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa-Holm equation, Kolmogorov-Petrovskii-Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the (2+1)-dimensional asymmetric version of the Nizhnik-Novikov-Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.     

A comparison result for radial solutions of the mean curvature equation

We establish two comparison results, between the solutions of a class of mean curvature equations and pieces of arcs of circles that satisfy the same Neumann boundary condition. Finally we present a number of examples where our estimates can be applied; some of them have a physical motivation.