(2+1)维Boussinesq方程的Backlund变换与精确解

借助于符号计算软件Maple，对方程的种子解作适当的未知函数替换，然后利用Backlund 变 换通过具体的符号演算获得了(2+1)维Boussinesq方程的一系列精确解.这些解包括类孤子解 和有理解,其中有的解中含有任意函数，当任意函数取特殊函数时，这些解具有丰富的结构 ，有些结构可能对物理现象的研究是有意义的. 关键词： (2+1)维Boussinesq方程 Backlund 变换 精确解 类孤子解     

Exact periodic waves and their interactions for the (2+1)-dimensional KdV equation

By means of the singular manifold method we obtain a general solution involving three arbitrary functions for the (2+1)-dimensional KdV equation. Diverse periodic wave solutions may be produced by appropriately selecting these arbitrary functions as the Jacobi elliptic functions. The interaction properties of the periodic waves are investigated numerically and found to be nonelastic. The long wave limit yields some new types of solitary wave solutions. Especially the dromion and the solitoff solutions obtained in this paper possess new types of solution structures which are quite different from the basic dromion and solitoff ones reported previously in the literature.     

On New Similarity Solutions of the Boiti–Leon–Pempinelli System

In the present work, the new exact solutions of the Boiti-Leon-Pempinelli system have been found. The system has extensive physical background. The exact solutions of the Boiti-Leon-Pempinelli system are investigated using similarity transformation method via Lie group theory. Lie symmetry generators are used for constructing similarity variables for the given system of partial differential equations, which lead to the new system of partial differentiaJ equations with one variable less at each step and eventually to a system of ordinary differential equations （ODEs）. Finally, these ODEs are solved exactly. The exact solutions are obtained under some parametric restrictions. The elastic behavior of the soliton solutions is shown graphically by taking some appropriate choices of the arbitrary functions involved in the solutions.     

THE EXACT SOLUTIONS OF THE BURGERS EQUATION AND HIGHER-ORDER BURGERS EQUATION IN (2+1) DIMENSIONS

Some exact solutions of the Burgers equation and higher-order Burgers equation in (2+1) dimensions are obtained by using the extended homogeneous balance method. In these solutions there are solitary wave solutions, close formal solutions for the initial value problems of the Burgers equation and higher-order Burgers equation, and also infinitely many rational function solutions. All of the solutions contain some arbitrary functions that may be related to the symmetry properties of the Burgers equation and the higher-order Burgers equation in (2+1) dimensions.     

Extended Mapping Transformation Method and Its Applications to Nonlinear Partial Differential Equation（s）

In this paper, we extend the mapping transformation method through introducing variable coefficients. By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.     

Extended Mapping Transformation Method and Its Applications to Nonlinear Partial Differential Equation(s)

In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.     

Primary Branch Solutions of First Order Autonomous Scalar Partial Differential Equations via Lie Symmetry Approach

A primary branch solution (PBS) is defined as a solution with m independent n ? 1 dimensional arbitrary functions for an m order n dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be determined by using Lie symmetry group approach companying with the introduction of auxiliary fields. Because of the intrusion of the arbitrary function, the PBSs have abundant and complicated structure. Usually, PBSs are implicit solutions. In some special cases, explicit solutions such as the instanton (rogue wave like) solutions may be obtained by suitably fixing the arbitrary function of the PBS.     

Chaotic solutions of (2+1)-dimensional Broek Kaup equation with variable coefficients

In this paper,an improved projective approach is used to obtain the variable separation solutions with two arbitrary functions of the (2+1)-dimensional Broek-Kaup equation with variable coefficients (VCBK). Based on the derived solitary wave solution and using a known chaotic system,some novel chaotic solutions are investigated.     

New Exact Solutions of （1 ＋ 1）-Dimensional Coupled Integrable Dispersionless System

This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more generalvariable separation solutions with two arbitrary functions of (1+1)-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p(x) and q(t) inour solutions, the annihilation phenomena of the flat-basin soliton, arch-basin soliton, and flat-top soliton are discussed.     

关于分离变量法的一个注记

从B-cklund变换出发，利用Cole-Hopf变换，一些有重要物理意义的(2+1)维非线性演化方程(组)被简化为具有两个分别关于(x,t),(y,t)的任意函数的单个线性偏微分方程.通过对该方程解的研究，获得了原方程一些新的精确解.其中，一些近年来被广泛研究的由分离变量法所获得的解也被重新得到. 关键词： B-cklund变换 Cole-Hopf变换 分离变量法 精确解     

Two Classes of New Exact Solutions to (2+1)-Dimensional Breaking Soliton Equation

The singular manifold method is used to obtain two general solutions to a (2 1)-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of the Jacobi elliptic functions are generated from the general solutions. The long wave limit yields the new types of dromion and solitary structures.     

Exact Solutions of (2+1)-Dimensional Boiti-Leon-Pempinelle Equation with (G'/G)-Expansion Method

In this paper, we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation by using the (G'/G)-expansion method, and with the help of Maple. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. This method can beapplied to other higher-dimensional nonlinear partial differential equations.     

Controllable Discrete Rogue Wave Solutions of the Ablowitz-Ladik Equation in Optics

With the aid of symbolic computation Maple, the discrete Ablowitz-Ladik equation is studied via an algebra method, some new rational solutions with four arbitrary parameters are constructed. By analyzing related parameters, the discrete rogue wave solutions with alterable positions and amplitude for the focusing Ablowitz-Ladik equations are derived. Some properties are discussed by graphical analysis, which might be helpful for understanding physical phenomena in optics.     

A General Method for Obtaining Degenerate Solutions to the Dirac and Weyl Equations and a Discussion on the Experimental Detection of Degenerate States

In this work, a general method is described for obtaining degenerate solutions of the Dirac equation, corresponding to an infinite number of electromagnetic 4-potentials and fields, which are explicitly calculated. More specifically, using four arbitrary real functions, one can automatically construct a spinor that satisfies the Dirac equation for an infinite number of electromagnetic 4-potentials, defined by those functions. An interesting characteristic of these solutions is that, in the case of Dirac particles with nonzero mass, the degenerate spinors should be localized, both in space and time. The method is also extended to the cases of massless Dirac and Weyl particles, where the localization of the spinors is no longer required. Finally, two experimental methods are proposed for detecting the presence of degenerate states.     

The Exact Solutions of Some (2+1)-Dimensional Integrable Systems

The equivalence of three (2+1)-dimensional soliton equations is proved, and the quite general solutions with some arbitrary functions of x, t and y respectively are obtained. By selecting the arbitrary functions, many special types of the localized excitations like the solitoff solitons, multi-dromion solutions, lump, and multi-ring soliton solutions are obtained.     

Double-Parameter Solutions of Projective Riccati Equations and Their Applications

The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values of coeffcients of PREs, the solutions with two arbitrary parameters of PREs can be expressed by the hyperbolic functions, the trigonometric functions, and the rational functions respectively, at the same time the relation between the components of each solution to PREs is also implemented. Second, more new travelling wave solutions for some nonlinear PDEs, such as the Burgers equation, the mKdV equation, the NLS^＋ equation, new Hamilton amplitude equation, and so on, are obtained by using Sub-ODE method, in which PREs are taken as the Sub-ODEs. The key idea of this method is that the travelling wave solutions of nonlinear PDE can be expressed by a polynomial in two variables, which are the components of each solution to PREs, provided that the homogeneous balance between the higher order derivatives and nonlinear terms in the equation is considered.     

Conditional Similarity Solutions of （2＋1）—Dimensional General nonintegrable KdV Equation

The real physics models are usually quite complex with some arbitrary parameters which will lead to the nonintegrability of the model.To find some exact solutions of a nonintegrable model with some arbitrary parameters is much more difficult than to find the solutions of a model with some special parameters.In this paper,we make a modification for the usual direct method to find some conditional similarity solutions of a (2 1)-dimensional general nonintegrable KdV equation.     

An Extended Subequation Rational Expansion Method and Solutions of (2+1)-Dimensional Cubic Nonlinear Schrödinger Equation

An extended subequation rational expansion method is presented and used to construct some exact analytical solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation. From our results, many known solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of new non-travelling wave and coefficient function's soliton-like solutions, and elliptic solutions are demonstrated by some plots.     

Several Types of Similarity Solutions for the Hunter-Saxton Equation

We study separable and self-similar solutions to the HunterSaxton equation,a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal(among other applications).Essentially,we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the HunterSaxton equation.For each type of solution,we are able to obtain some simple exact solutions in closed-form,and more complicated solutions through an analytical approach.We find that there is a whole family of self-similar solutions,each of which depends on an arbitrary parameter.This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data.The simpler solutions found constitute exact solutions to a nonlinear partial differential equation,and hence are also useful in a mathematical sense.Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions.Both types of solutions cast light on self-similar phenomenon arising in the HunterSaxton equation.     

New Exact Solutions and Special Coherent Structures for Coupled Integrable Dispersionless Equation

Using improved homogeneous balance method, we obtain new exact solutions for the coupled integrable dispersionless equation. On the basis of these exact solutions, we find some new interesting coherent structures by selecting arbitrary functions appropriately.