Dynamics of D’Alembert wave and soliton molecule for a (2+1)-dimensional generalized breaking soliton equation

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.     

Soliton-Like Solutions for the (2+l)-Dimensional Nonlinear Evolution Equation

By using a homogeneous balance method, we give new soliton-like solutions for the (2+1)-dimensional KdV equation and the (2+1)-dimensional breaking soliton equation. Solitary wave soIutions are shown to be a special case of the present results.     

(2+1)维破裂孤子方程的多方孤子解及其混沌行为

利用投射方程法和变量分离法,得到(2+1)维破裂孤子方程的新显式精确解. 根据得到的孤立波解,利用 Weierstrassp 函数,构造出多方孤子局域结构. 利用两个混沌系统研究了破裂孤子方程的混沌行为. 关键词： 投射方程法 破裂孤子方程 多方孤子 混沌行为     

Symbolic Computation and Construction of Soliton-Like Solutions to the(2+1)-Dimensional Breaking Soliton Equation

Based on the computerized symbolic system Maple, a new generalized expansion method of Riccatiequation for constructing non-travelling wave and coefficient functions‘ soliton-like solutions is presented by a new generalansatz. Making use of the method, we consider the (2 1)-dimensional breaking soliton equation, ut buxxy 4buvx 4buxv = 0, uy = vx, and obtain rich new families of the exact solutions of the breaking soliton equation, including thenon-travelling wave and constant function soliton-like solutions, singular soliton-like solutions, and triangular functionsolutions.     

Reduction and New Explicit Solutions of (2+1)-Dimensional Breaking Soliton Equation

By applying the Lie group method, the (2+1)-dimensional breaking soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional breaking soliton equation are obtained.     

Symbolic Computation and Construction of Soliton－Like Solutions to the（2＋l）－Dimensional Breaking Soliton Equation

Based on the computerized symbolic system Mapte, a new generalized expansion method of Riccati equation for constructing non-travelling wave and coefficient functions‘ soliton-like solutions is presented by a new general ansatz. Making use of the method, we consider the (2 1)-dimensional breaking soliton equation, ut buxxy 4buvx 4buxv = O,uv=vx, and obtain rich new families of the exact solutions of the breaking sofiton equation, including then on-traveilin~ wave and constant function sofiton-like solutions, singular soliton-like solutions, and triangular function solutions.     

Symmetry Reduction and Exact Solutions of the (3+1)-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the (3+1)-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the (3+1)-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the (3+1)-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the (3+1)-dimensional breaking soliton equation.     

New Double-Periodic Soliton Solutions for the (2+1)-Dimensional Breaking Soliton Equation

Under investigation is the (2+1)-dimensional breaking soliton equation. Based on a special ansätz functions and the bilinear form, some entirely new double-periodic soliton solutions for the (2+1)-dimensional breaking soliton equation are presented. With the help of symbolic computation software Mathematica, many important and interesting properties for these obtained solutions are revealed with some figures.     

Double Elliptic Equation Expansion Approach and Novel Solutions of （2＋1）-Dimensional Break Soliton Equation

In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the （2＋1）-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in （2＋1） dimensions.     

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 breaking soliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutions and triangular periodic wave solutions are obtained.     

(2+1)维破裂孤子方程的多 Solitoff 解及其演化

借助计算机 Maple 软件系统,利用拓展的(G'/G)方法和变量分离方法, 得到(2+1)维破裂孤子方程的精确解. 根据得到的孤立波解, 构造出多 Solitoff 局域结构, 研究了孤子随时间的演化.     

Abundant Multisoliton Structure of (3+1)-Dimensional Breaking Soliton Equation

Using the extended homogeneous balance method, we obtained abundant exact solution structures of the (3 1 )-dimensional breaking soliton equation. By means of the leading order term analysis, the nonlinear transformations of the (3t1)-dimensional breaking soliton equation are given first, and then some special types of single solitary wave solutions and the multisoliton solutions are constructed.     

New Generalized Transformation Method and Its Application in Higher-Dimensional Soliton Equation

A new generalized transformation method is presented to find more exact solutions of nonlinear partial differential equation. As an application of the method, we choose the (3+1)-dimensional breaking soliton equation to illustrate the method. As a result many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic function solutions, and rational solutions, are obtained. The new method can be extended to other nonlinear partial differential equations in mathematical physics.     

Lie Symmetries,Infinite-Dimensional Lie Algebras and Similarity Reductions of Certain (2+1)-Dimensional Nonlinear Evolution Equations

Abstract

The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik – Novikov – Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation, the (2+1)-dimensional generalization of the nonlinear Schrödinger equation by Fokas as well as the (2+1)-dimensional generalized sine-Gordon equation of Konopelchenko and Rogers. We show that in all these cases the Lie symmetry algebra is infinite-dimensional; however, in the case of the breaking soliton equation they do not possess a centerless Virasorotype subalgebra as in the case of other typical integrable (2+1)-dimensional evolution equations. We work out the similarity variables and special similarity reductions and investigate them.     

Construction of Soliton-Cnoidal Wave Interaction Solution for the (2+1)-Dimensional Breaking Soliton Equation

In this paper, the truncated Painlev′e analysis and the consistent tanh expansion(CTE) method are developed for the(2+1)-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is difficult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus m = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.     

Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

The (2 1)-dimensional nonlinear barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a beta-plane channel is investigated by using the classical Lie symmetry approach. Some types of group-invariant wave solutions are expressed by means of the lower-dimensional similarity reduction equations. In addition to the known periodic Rossby wave solutions, some new types of exact solutions such as the ring solitary waves and the breaking soliton type of vorticity solutions with nonlinear and nonconstant shears are also obtained.     

A New Integrable (2+1)-Dimensional Generalized Breaking Soliton Equation: N-Soliton Solutions and Traveling Wave Solutions

In this work, we study a new (2+1)-dimensional generalized breaking soliton equation which admits the Painleve property for one special set of parameters. We derive multiple soliton solutions, traveling wave solutions, and periodic solutions as well. We use the simplified Hirotas method and a variety of ansatze to achieve our goal.     

扩展的形变映射方法和(2+1)维破裂孤子方程的新解

扩展的形变映射方法应用于非线性物理模型的研究， 获得了(2+1)维破裂孤子方程的新解, 合适地选择任意函数，可以获得该模型的丰富的局域和非局域的相干结构，这里仅揭示(2+1)维破裂孤子方程(1)和(2)的周期型新孤波结构. 关键词： 形变映射方法 (2+1)维破裂孤子方程 孤波结构     

CONDITIONAL SIMILARITY REDUCTION APPROACH: JIMBO－MIWA EQUATION

The direct method developed by Clarkson and Kruskal (1989 J. Math. Phys. 30 2201) for finding the symmetry reductions of a nonlinear system is extended to find the conditional similarity solutions. Using the method of the Jimbo－Miwa (JM) equation, we find that three well-known (2+1)-dimensional models－the asymmetric Nizhnik--Novikov－Veselov equation, the breaking soliton equation and the Kadomtsev-Petviashvili equation－can all be obtained as the conditional similarity reductions of the JM equation.     

Bäcklund Transformations,Solitary Waves,Conoid Waves and Bessel Waves of the (2+1)-Dimensional Euler equation

Some simple special Bäcklund transformation theorems are proposed and utilized to obtain exact solutions for the (2+1)-dimensional Euler equation. It is found that the (2+1)-dimensional Euler equation possesses abundant soliton or solitary wave structures, conoid periodic wave structures and the quasi-periodic Bessel wave structures on account of the arbitrary functions in its solutions. Moreover, all solutions of the arbitrary two dimensional nonlinear Poisson equation can be used to construct exact solutions of the (2+1)-dimensional Euler equation.