Painlevé Analysis,Soliton Solutions and Bcklund Transformation for Extended (2 + 1)-Dimensional Konopelchenko-Dubrovsky Equations in Fluid Mechanics via Symbolic Computation

This paper is to investigate the extended(2+1)-dimensional Konopelchenko-Dubrovsky equations,which can be applied to describing certain phenomena in the stratified shear flow,the internal and shallow-water waves, plasmas and other fields.Painleve analysis is passed through via symbolic computation.Bilinear-form equations are constructed and soliton solutions are derived.Soliton solutions and interactions are illustrated.Bilinear-form Backlund transformation and a type of solutions are obtained.

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.

A New Class of Periodic Solutions to （2＋1）-Dimensional KdV Equations

We investigate a new class of periodic solutions to (2+1)-dimensional KdV equations, by both the linear superposition approach and the mapping deformation method. These new periodic solutions are suitable combinations of the periodic solutions to the (2+1)-dimensional KdV equations obtained by means of the Jacobian elliptic function method, but they possess different periods and velocities.

On an Auto-Bācklund 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-Bācklund 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.

New Families of Rational Form Variable Separation Solutions to(2+1)-Dimensional Dispersive Long Wave Equations

在符号的计算系统枫树的帮助下，新合理可变分离解决方案的一些家庭( 2+1 )维的散长波浪方程借助于函数转变，改进印射的途径，和可变分离途径，在之中有合理独居的波浪解决方案被构造，周期的波浪解决方案和合理波浪解决方案。

Symmetry Groups and New Exact Solutions of(2+1)-Dimensional Dispersive Long-Wave Equations

用修改 CK 的直接方法，我们导出一条对称组定理(2+1 ) 维的散长波浪的方程。在定理，谎言点对称组和新准确答案之上基于(2+1 ) 维的散长波浪的方程被获得。

New Exact Solutions for （2＋1）-Dimensional Breaking Soliton Equation

New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutionsand triangular periodic wave solutions are obtained.

The Travelling Wave Solutions for （2＋1）-dimensional AKNS Equation

Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic solution. It also proves that the result is consistent with the soliton solution of simplify Hirota bilinear method by Wazwaz and illustrate the solution are right travelling wave solution.

Exact Solutions to (2+1)-Dimensional Kaup-Kupershmidt Equation

In this paper,by using the symmetry method,the relationships between new explicit solutions and oldones of the(2+1)-dimensional Kaup-Kupershmidt(KK)equation are presented.We successfully obtain more generalexact travelhng wave solutions for(2+1)-dimensional KK equation by the symmetry method and the(G'/G)-expansionmethod.Consequently,we find some new solutions of(2+1)-dimensional KK equation,including similarity solutions,solitary wave solutions,and periodic solutions.

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.

Wronskian and Grammian Solutions for(2+1)-Dimensional Soliton Equation

在这份报纸，(2 + 1 ) 维的 soliton 方程主要正在被讨论。把直接方法， Wronskian 技术和 Pfaffian 基于 Hirota 性质， N-soliton 答案， Wronskian 和 Grammian 答案被产生了。

Exotic Localized Coherent Structures of New（2＋1）－Dimensional Soliton Equation

The variable separation approach is used to obtain localized coherent structures of the new (2+1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryfunctions of the seed solutions, the abundance of the localized structures of this model are derived. Some special types ofsolutions solitoff, dromions, dromion lattice, breathers and instantons are discussed by selecting the arbitrary functionsappropriately. The breathers may breath in their amplititudes, shapes, distances among the peaks and even the numberof the peaks.

Nonlocal symmetry and exact solutions of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation

In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the（2＋1）-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the（2＋1）-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion（CRE） solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.

New periodic wave solutions, localized excitations and their interaction for (2+1)-dimensional Burgers equation

Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for （2＋l）-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions （which contains two arbitrary functions）. Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave （called semi-localized） patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations （two-dromion solution）.

Algebraic-Geometric Solution to (2+1)-Dimensional Sawada-Kotera Equation

特殊答案(2+1 ) 维的 Sawada-Kotera 方程被分解成三(0+1 ) 维的 Bargmann 流动。他们在联系亢奋的椭圆形的曲线的 Jacobi 变化上被澄清。明确的代数学几何的答案根据 KdV 层次的更深的理解被获得。

The exact solutions to (2+1)-dimensional nonlinear Schrdinger equation

By using the extended F-expansion method, the exact solutions,including periodic wave solutions expressed by Jacobi elliptic functions, for (2+1)-dimensional nonlinear Schrdinger equation are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.

A New (2+1)-Dimensional KdV Equation and Its Localized Structures

A new (2+1)-dimensional KdV equation is constructed by using Lax pair generating technique.Exactsolutions of the new equation are studied by means of the singular manifold method.Bcklund transformation in termsof the singular manifold is obtained.And localized structures are also investigated.

Painlevé Property and Exact Solutions to a (2+1) Dimensional KdV-mKdV Equation

A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.