Two new discrete integrable systems

In this paper,we focus on the construction of new(1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new(1+1)-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.     

Multisoliton Solutions of the (2+1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2+1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2+1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.     

On an Auto-Bäcklund Transformation for (2+1)-Dimensional Variable Coefficient Generalized KP Equations and Exact Solutions

By the application of the extended homogeneous balance method, we derive an auto-Bäcklund transformation (BT) for (2+1)-dimensional variable coefficient generalized KP equations. Based on the BT, in which there are two homogeneity equations to be solved, we obtain some exact solutions containing single solitary waves.     

Stair and Step Soliton Solutions of the Integrable (2+1) and (3+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equations

The multiple exp-function method is a new approach to obtain multiple wave solutions of nonlinear partial differential equations (NLPDEs). By this method one can obtain multi-soliton solutions of NLPDEs. In this paper, using computer algebra systems, we apply the multiple exp-function method to construct the exact multiple wave solutions of a (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Also, we extend the equation to a (3+1)-dimensional case and obtain some exact solutions for the new equation by applying the multiple exp-function method. By these applications, we obtain single-wave, double-wave and multi-wave solutions for these equations.     

Stair and Step Soliton Solutions of the Integrable (2+1) and (3+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equations

The multiple exp-function method is a new approach to obtain multiple wave solutions of nonlinear partial differential equations (NLPDEs). By this method one can obtain multi-soliton solutions of NLPDEs. In this paper, using computer algebra systems, we apply the multiple exp-function method to construct the exact multiple wave solutions of a (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Also, we extend the equation to a (3+1)-dimensional case and obtain some exact solutions for the new equation by applying the multiple exp-function method. By these applications, we obtain single-wave, double-wave and multi-wave solutions for these equations.     

New exact periodic solutions to （2＋1）-dimensional dispersive long wave equations

In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.     

(3+1)维非线性Burgers系统的新的分离变量解及其局域激发结构与分形结构

将扩展的Riccati方程映射法推广到了(3+1)维非线性Burgers系统，得到了系统的分离变量解；由于在解中含有一个关于自变量(x,y,z,t)的任意函数，通过对这个任意函数的适当选取，并借助于数学软件Mathematica进行数值模拟，得到了系统的新而丰富的局域激发结构和分形结构.结果表明，扩展的Riccati方程映射法在求解高维非线性系统时，仍然是一种行之有效的方法，并且可以得到比(2+1)维非线性系统更为丰富的局域激发结构. 关键词： 扩展的Riccati方程映射法 (3+1)维非线性Burgers方程 局域激发结构 分形结构     

New Explicit Exact Solutions to (2+1)-Dimensional Generalized Broer-Kaup System

A nonlinear transformation and some multi-solition solutions for the (2+1)-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear transformation, we reduce the (2+1)-dimensional GBK system to a simple linear evolution equation. Solving this equation, we can obtain some new explicit exact solutions of the original equations by means of the extended hyperbola function method.     

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     

Solving Integrable Broer-Kaup Equations in （2＋1）-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Baecklund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x, t} and {y, t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding(2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation(BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing(2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the(2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the(2+1)-dimensional AKNS equation(also called the Davey-Stewartson hierarchy), a kind of(2+1)-dimensional Schr¨odinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new(2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the(2+1)-dimensional integrable coupling, which is further reduced to the standard(2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known(1+1)-dimensional AKNS hierarchy, the(1+1)-dimensional nonlinear Schr¨odinger equation are all special cases of the(2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the(2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.     

（2+1）维耗散长波方程与(2+1)维Broer-Kaup方程新的类多孤子解

进一步拓广齐次平衡法的应用，并对关键的操作步骤进行了改进，从而简便地求出了（2+1 ）维耗散长波方程和（2+1）维Broer-Kaup方程新的类多孤子解-这种解更具有一般性，它包 含着已有文献给出的类多孤子解- 关键词： 齐次平衡法 类多孤子解 （2+1）维耗散长波方程 （2+1）维Broer-Kaup方程     

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A₁, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, in-cluding the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.     

（2＋2）-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.     

Multisoliton Solutions of the (2 1)-Dimensional KdV Equation

Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2 1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2 1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.``     

Similarity Reductions of Nonisospectral KP Equation by a Direct Method

On bases of the direct method developed by Clarkson and Kruskal [J. Math. Phys. 27 (1989) 2201], the (2+1)-dimensional nonisospectral Kadomtsev-Petviashvili (KP) equation has been reduced to three types of (1+1)-dimensional partial differential equations. We focus on solving the third type of reduction and dividing them into three subcases, from which we obtain rich solutions including some arbitrary functions.     

Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.     

Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations

We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations, As concrete examples of its application, we apply this method to the （2＋1）-dimensional modified Broer- Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.     

Discrete integrable couplings associated with Toda-type lattice and two hierarchies of discrete soliton equations

With the help of two semi-direct sum Lie algebras, an efficient way to construct discrete integrable couplings is proposed. As its applications, the discrete integrable couplings of the Toda-type lattice equations are obtained. The approach can be devoted to establishing other discrete integrable couplings of the discrete lattice integrable hierarchies of evolution equations.     

New Bilinear Bäcklund Transformation and Higher Order Rogue Waves with Controllable Center of a Generalized (3+1)-Dimensional Nonlinear Wave Equation

In this paper, we first obtain a bilinear form with small perturbation u_0 for a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear B?cklund transformation which consists of four bilinear equations and involves seven arbitrary parameters is constructed. After that, by applying a new symbolic computation method, we construct the higher order rogue waves with controllable center to the generalized(3+1)-dimensional nonlinear wave equation. The rogue waves present new structure, which contain two free parametersα and β. The dynamic properties of the higher order rogue waves are demonstrated graphically. The graphs tell that the parameters α and β can control the center of the rogue waves.