应用exp-函数法求(n+1)维sine-Gordon方程的孤立波解

应用exp—函数法求得(n+1)维sine-Gordon方程的单孤子解、双孤子解、三孤子解,通过选取适当的参数,分别做出了单孤子解、双孤子解、三孤子解的函数图像,刻画了解的结构和性质.实践证明,应用exp-函数法研究非线性偏微分方程具有十分重要的作用和意义.     

Integrable (2+1)-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The Gibbons-Tsarev (GT) systems are most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g = 0 and g = 1 and also a new GT system corresponding to algebraic curves of genus g = 2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is “trivial.”     

On a 2 + 1-dimensional Darboux system: Integrable and geometric connections

It is shown that a novel 2 + 1-dimensional system recently introduced by Konopelchenko and Rogers contains as a specialization the Zakharov-Manakov matrix triad system. The latter, in turn, in its scalar version yields a classical system investigated by Darboux in connection with conjugate coordinate systems. This Darboux system, in a 1 + 1-dimensional reduction, turns out to be connected to the self-induced transparency equations. Here, geometric aspects of the 2 + 1-dimensional Darboux systems are recorded.     

Integrable decompositions for the (2+1)-dimensional Gardner equation

In this paper, with the computerized symbolic computation, the nonlinearization technique of Lax pairs is applied to find the integrable decompositions for the (2+1)-dimensional Gardner [(2+1)-DG] equation. First, the mono-nonlinearization leads a single Lax pair of the (2+1)-DG equation to a generalized Burgers hierarchy which is linearizable via the Hopf–Cole transformation. Second, by the binary nonlinearization of two symmetry Lax pairs, the (2+1)-DG equation is decomposed into the generalized coupled mixed derivative nonlinear Schrödinger (CMDNLS) system and its third-order extension. Furthermore, the Lax representation and Darboux transformation for the CMDNLS and third-order CMDNLS systems are constructed. Based on the two integrable decompositions, the resonant N-shock-wave solution and an upside-down bell-shaped solitary-wave solution are obtained and the relevant propagation characteristics are discussed through the graphical analysis.     

On the kinematics of 2+1-dimensional motions of a fibre-reinforced fluid. Integrable connections

Evolution of foliations of the plane is shown, under a conditionof constant divergence, to be linked to the scattering problemfor the integrable modified Korteweg–de Vries hierarchy.This result is applied to a set of kinematic relations whicharise in the theory of ideal fibre-reinforced fluids. In particular,it is established that the fibres, which are convected withthe fluid, constitute generalized tractrices.     

一个2+1维可积方程的代数几何解

该文从1+1维的孤子方程出发,构造出一个2+1维在Lax意义下可积的方程.接着这个2+1维可积方程被分解为可解的常微分方程.随后引入超椭圆Riemann曲面和Abel-Jacobi坐标把流进行了拉直.再利用Riemannθ函数给出了这个2+1维方程的代数几何解.     

Integrable boundary conditions for (2+1)-dimensional models of mathematical physics

We consider the question of integrable boundary-value problems in the examples of the two-dimensional Toda chain and Kadomtsev-Petviashvili equation. We discuss the problems that are integrable from the standpoints of two basic definitions of integrability. As a result, we propose a method for constructing a hierarchy of integrable boundary-value problems where the boundaries are cylindric surfaces in the space of three variables. We write explicit formulas describing wide classes of solutions of these boundary-value problems for the two-dimensional Toda chain and Kadomtsev-Petviashvili equation.     

New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine-Gordon equation

New exact solutions including the kink-shaped solutions, bell-shaped solutions, periodic solutions, singular soliton solutions and rational solution for the (2 + 1)-dimensional Sine-Gordon equation are obtained using the tanh method, the -expansion method and the auxiliary function method, respectively.     

(2+1)维Burgers方程新的复合解

In this paper, a new generalized compound Riccati equations rational expansion method （GCRERE） is proposed. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method is not only recover some known solutions, but also find some new and general complexiton solutions. Being concise and straightforward, it is applied to the （2＋1）-dimensional Burgers equation. As a result, eight families of new exact analytical solutions for this equation are found. The method can also be applied to other nonlinear partial differential equations.     

A Numerical Study of the Light Bullets Interaction in the (2+1) Sine-Gordon Equation

The propagation and interaction in more than one space dimension of localized pulse solutions (so-called light bullets) to the sine-Gordon [SG] equation is studied both asymptotically and numerically. Similar solutions and their resemblance to solitons in integrable systems were observed numerically before in vector Maxwell systems. The simplicity of SG allows us to perform an asymptotic analysis of counterpropagating pulses, as well as a fully resolved computation over rectangular domains. Numerical experiments are carried out on single pulse propagation and on two pulse collision under different orientations. The particle nature, as known for solitons, persists in these two space dimensional solutions as long as the amplitudes of initial data range in a finite interval, similar to the conditions on the vector Maxwell systems.     

(3+1)维extended Jimbo-Miwa方程的精确解

应用Hirota双线性方法和新三波测试函数研究了(3+1)维extend JimboMiwa方程,借助Maple计算软件,获得了12组它的精确解.通过选取适当的参数,做出一部分精确解的图形来说明他们的物理特征.     

利用Tanh方法产生的(2+1)和(3+1)维非线性可积方程的精确解

主要利用Tanh函数方法,对两个高维五阶非线性可积方程进行了讨论,通过行波约化,分别将(2+1)和(3+1)维非线性可积方程转化为常微分方程.结合Riccati方程的性质,分别得到关于若干参变量的代数系统,借助于Mathematica软件符号运算功能,最终得到了上述两个高维方程的精确解.     

广义(2+1)维Boussinesq方程的孤立波解

首先应用Riccati展开法获得广义(2+1)维Boussinesq方程的96组相互作用解,这类解同时含有三角函数、双曲函数、有理函数、指数函数等,它反映了不同类型非线性波的相互作用.然后应用同宿测试方法结合Hirota双线性形式求得广义(2+1)维Boussinesq方程的周期孤波解,通过相应的时空变换,得到方程其他形式的解.     

One variant of a (2 + 1)-dimensional Volterra system and its (1 + 1)-dimensional reduction

A new system is generated from a multi-linear form of a (2+1)-dimensional Volterra system. Though the system is only partially integrable and needs additional conditions to possess two-soliton solutions, its (1+1)-dimensional reduction gives an integrable equation which has been studied via reduction skills. Here, we give this (1+1)-dimensional reduction a simple bilinear form, from which a Bäcklund transformation is derived and the corresponding nonlinear superposition formula is built.     

Neumann边界条件下离散化强阻尼Sine-Gordon方程的全局吸引性

本文主要研究Neumann边界条件下带有周期外力的离散化强阻尼Sine-Gordon方程的全局吸引性.当摩擦系数足够大时,该系统将拥有一族周期解,这些周期解彼此之间只相差-个常向量的整数倍,并且该系统的任意一个解都将被其中的-个周期解所吸引.     

Symmetries in (1+1)-dimensional Integrable System with Their Algebraic Structures and Conserved Quantities

In this paper we offer a general method or constructing symmetries and conserved quantities in the (1 + 1)-dimensional integrable system, prove the algebraic relations between symmetries, and what is more, give applications of this method in many integrable systems with physical significance.     

（２＋１）维非线性色散长波方程的相似约化和解析解

首先借助于Ｍａｔｈｅｍａｔｉｃａ软件,将Ｃｌａｒｋｓｏｎ和Ｋｒｕｓｋａｌ引入的直接约化法推广并应用于（２＋１） 维偏微分方程组情形 （２＋１） 维非线性色散长波方程,获得了该方程的六种类型的相似约化和若干解析解,其中包括ＰａｉｎｌｅｖｅⅡ型方程和孤子解．然后基于文［５］的结论,通过引入新的级数变换,获得了该方程的有理分式解析解．这种方法也适合于其它的微分方程．