Novel composite function solutions of the modified KdV equation

A Wronskian form expansion method is proposed to construct novel composite function solutions to the modified Korteweg-de Vries (mKdV) equation. The method takes advantage of the forms and structures of Wronskian solutions to the mKdV equation, and Wronskian entries do not satisfy linear partial differential equations. The method can be automatically carried out in computer algebra (for example, Maple).     

2+1 维变系数广义KP方程的椭圆周期解

运用Jacobi椭圆函数展开法求得了2 1维变系数广义KadoratsevPetviashvili方程的椭圆周期解及孤立波解．     

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied 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.     

On Weierstrass elliptic function solutions for a (N+1) dimensional potential KdV equation

This paper is concerned with several aspects of travelling wave solutions for a (N+1) dimensional potential KdV equation. The Weierstrass elliptic function solutions, the Jaccobi elliptic function solutions, solitary wave solutions, periodic wave solutions to the equation are acquired under certain circumstances. It is shown that the coefficients of derivative terms in the equation cause the qualitative changes of physical structures of the solutions.     

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.     

Analytical solutions to a generalized Drinfel’d-Sokolov equation related to DSSH and KdV6

Analytical solutions to the generalized Drinfel’d-Sokolov (GDS) equations

     

一般变系数KdV方程的精确解

By asing the nonclassical method of symmetry reductions, the exact solutions for general variable-coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don‘t exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable-coefficient KdV equation is given.     

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.     

Numerical solutions of KdV equation using radial basis functions

Numerical solution of the Korteweg-de Vries equation is obtained by using the meshless method based on the collocation with radial basis functions. Five standard radial basis functions are used in the method of the collocation. The results are compared for the numerical experiments of the propagation of solitons, interaction of two solitary waves and breakdown of initial conditions into a train of solitons.     

High order nonlocal symmetries and exact interaction solutions of the variable coefficient KdV equation

In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior.     

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.     

NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF INHOMOGENEOUS p-LAPLACE EQUATION

LetΩbe a smooth bounded domain in Rn. In this article, we consider the homogeneous boundary Dirichlet problem of inhomogeneous p-Laplace equation -△pu=|u|q-1u λf(x) onΩ, and identify necessary and sufficient conditions onΩand f(x) which ensure the existence, or multiplicities of nonnegative solutions for the problem under consideration.     

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 gene...     

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.     

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.     

Solitary wave solutions of the improved KdV equation by VIM

The variational iteration method (VIM) is applied to solve numerically the improved Korteweg-de Vries equation (IKdV). A correction function is constructed with a general Lagrange multiplier that can be identified optimally via the variational theory. This technique provides a sequence of functions with easily computable components that converge rapidly to the exact solution of the IKdV equation. Propagation of single, interaction of two, and three solitary waves, and also birth of solitons have been discussed. Three invariants of motion have been evaluated to determine the conservation properties of the problem. This procedure is promising for solving other nonlinear equations.     

A series of the solutions for the Heisenberg ferromagnetic spin chain equation

The Heisenberg ferromagnetic spin chain equation is investigated. By applying the improved F‐expansion method (Exp‐function method) and the Jacobi elliptic method, respectively, a series of exact solutions is constructed. The parametric conditions of the existence for the solutions are presented. These solutions comprise periodic wave solutions, doubly periodic wave solutions, and dark and bright soliton solutions, which are expressed in several different function forms, namely, Jacobi elliptic function, trigonometric function, hyperbolic function, and exponential function. The results illustrate that the Exp‐function method is a powerful symbolic algorithm to look for new solutions for the nonlinear evolution systems.     

高阶KdV方程的复化解结构

通过行波变换将高阶KdV方程转换成复域中的常微分方程,以Nevanlinna值分布理论的有关知识为基础,研究了复化的高阶KdV方程w(4)+w″+1/2w2-c2-b=0(其中c,b为复常数)的亚纯解结构,确定了可能的三种形式的亚纯解.对于两类高阶方程(nKdV)1和(mKdV)2,当n=2,3和m=3时,不能确定相应的复化方程有类似亚纯解结构;当m=2时,相应复化方程具有具体形式的亚纯解.     

New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation

In this work, a new generalized Jacobi elliptic functions expansion method based upon four new Jacobi elliptic functions is described and abundant new Jacobi-like elliptic functions solutions for the variable-coefficient mKdV equation are obtained by using this method, some of these solutions are degenerated to solitary-like solutions and triangular-like functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m→1 or 0, which shows that the new method can be also used to solve other nonlinear partial differential equations in mathematical physics.