Interactions among special embed-solitons for the (3+1)-dimensional Burgers equation

With the help of a modified mapping method and a new mapping method, we re-study the (3+1)-dimensional Burgers equation, and derive two families of variable separation solutions. By selecting appropriate functions in the variable separation solution, we discuss the interaction behaviors among taper-like, plateau-type rings, and rectangle-type embed-solitons in the periodic wave background. All the interaction behaviors are completely elastic, and no phase shift appears after interaction.     

A New Class of (2+1)-Dimensional Localized Coherent Structures with Completely Elastic and Non-elastic Interactive Properties

From the variable separation solution and by selecting appropriate functions, a new class of localized coherent structures consisting of solitons in various types are found in the (2 1)-dimensional long-wave-short-wave resonance interaction equation. The completely elastic and non-elastic interactive behavior between the dromion and compacton, dromion and peakon, as well as between peakon and compacton are investigated. The novel features exhibited by these new structures are revealed for the first time.     

A New Class of （2＋1）-Dimensional Localized Coherent Structures with CompletelyElastic and Non-elastic Interactive Properties

From the variable separation solution and by selecting appropriate functions, a new class of localized coherent structures consisting of solitons in various types are found in the (2 1)-dimensional long-wave-short-wave resonance interaction equation. The completely elastic and non-elastic interactive behavior between the dromion and compacton, dromion and peakon, as well as between peakon and compacton are investigated. The novel features exhibited by these new structures are revealed for the first time.     

Non-completely elastic interactions in a （2＋1）-dimensional dispersive long wave equation

With the help of a modified mapping method,we obtain two kinds of variable separation solutions with two arbitrary functions for the(2+1)-dimensional dispersive long wave equation.When selecting appropriate multi-valued functions in the variable separation solution,we investigate the interactions among special multi-dromions,dromion-like multi-peakons,and dromion-like multi-semifoldons,which all demonstrate non-completely elastic properties.     

Variable Separation Solutions of Generalized Broer-Kaup System via a Projective Method

Using an extended projective method, a new type of variable separation solution with two arbitrary functions of the (2+1)-dimensional generalized Broer-Kaup system (GBK) is derived. Based on the derived variable separation solution, some special localized coherent soliton excitations with or without elastic behaviors such as dromions, peakons, and foldons etc. are revealed by selecting appropriate functions in this paper.     

New Periodic Wave Solutions, Localized Excitations and Their Interaction Properties for （3＋1）-Dimensional Jimbo-Miwa Equation

Based on the multi-linear variable separation approach, a class of exact, doubly periodic wave solutions for the （3＋1）-dimensional Jimbo-Miwa equation is analytically obtained by choosing the Jacobi elliptic functions and their combinations. Limit cases are considered and some new solitary structures （new dromions） are derived. The interaction properties of periodic waves are numerically studied and found to be inelastic. Under long wave limit, two sets of new solution structures （dromions） are given. The interaction properties of these solutions reveal that some of them are completely elastic and some are inelastic.     

THE NON-LINEAR DYNAMIC BEHAVIOR OF AN ELASTIC LINKAGE MECHANISM WITH CLEARANCES

In terms of Newton two-state model, by choosing two sets of generalized co-ordinates, this paper develops a unified dynamic model between the separation and collision process for the elastic linkage mechanism. This model incorporates the effects of rigidity and elasticity coupling and the angular velocity of crank is assumed to be variable in the operation. In addition, this paper provides a more simple and practical numerical solution method for convenient analysis. Through an example, the dynamic responses of the elastic linkage mechanism with clearances are analyzed, both the effects of elasticity and clearance on the dynamic behaviors of the mechanism are analyzed simultaneously and the non-linear behaviors caused by the clearance joints are analyzed by the dynamic model of rigid mechanism.     

Interaction between compacton and anticompacton,peakon and antipeakon in （2 ＋ 1）-dimensional spaces

Starting from the variable separation solution obtained by using the extended homogenous balance method, a class of novel localized coherent structures such as the multi-peakon－antipeakons solution and the multi-compacton－anticompactons solution of the (2+1)-dimensional dispersive long wave equation are found by selecting appropriate functions. These new structures exhibit some novel interaction features that are different from one of the known results. Their interaction behaviour is very similar to the completely elastic collisions between two classical particles.     

(2+1)-Dimensional Combined Structures of Various Solitons with Completely Nonelastic Interaction Properties

Starting from the variable separation solution obtained by using the extended homogenous balance method, a new class of combined structures, such as multi-peakon and multi-dromion solution, multi-compacton and multi-dromion solution, multi-peakon and multi-compacton solution, for the (2+1)-dimensional Nizhnik-Novikov-Veselov equation are found by selecting appropriate functions. These new structures exhibit novel interaction features. Their interaction behavior is very similar to the completely nonelastic collisions between two classical particles.     

Solitons in a generalized (2+1)-dimensional Ablowitz－Kaup－Newell－Segur system

In the previous Letter (Zheng C L and Zhang J F 2002 Chin. Phys. Lett. 19 1399), a localized excitation of the generalized Ablowitz－Kaup－Newell－Segur (GAKNS) system was obtained via the standard Painlevé truncated expansion and a special variable separation approach. In this work, starting from a new variable separation approach, a more general variable separation excitation of this system is derived. The abundance of the localized coherent soliton excitations like dromions, lumps, rings, peakons and oscillating soliton excitations can be constructed by introducing appropriate lower-dimensional soliton patterns. Meanwhile we discuss two kinds of interactions of solitons. One is the interaction between the travelling peakon type soliton excitations, which is not completely elastic. The other is the interaction between the travelling ring type soliton excitations, which is completely elastic.     

New localized excitations and cross-like fractal structures to the (2+1)-dimensional Broer–Kaup system

A broad general variable separation solution with two arbitrary lower-dimensional functions of the (2+1)-dimensional Broer–Kaup (BK) equations was derived by means of a projective equation method and a variable separation hypothesis. Based on the derived variable separation excitation, some new special types of localized solutions such as oscillating solitons, instanton-like and cross-like fractal structures are revealed by selecting appropriate functions of the general variable separation solution.     

A Variable Separation Approach to Solve the Integrable and Nonintegrable Models:Coherent Structures of the (2 + 1)-Dimensional KdV Eqnation

We study the localized coherent structures ofa generally nonintegrable (2 1 )-dimensional KdV equation via a variable separation approach. In a special integrable case, the entrance of some arbitrary functions leads to abundant coherent structures. However, in the general nonintegrable case, an additional condition has to be introduced for these arbitrary functions. Although the additional condition has been introduced into the solutions of the nonintegrable KdV equation, there still exist many interesting solitary wave structures. Especially, the nonintegrable KdV equation possesses the breather-like localized excitations, and the similar static ring soliton solutions as in the integrable case. Furthermor,in the integrable case, the interaction between two travelling ring solitons is elastic, while in the nonintegrable case we cannot find even the single travelling ring soliton solution.     

Periodic Semifolded Solitary Waves for （2＋1）-Dimensional Variable Coefficient Broer-Kaup System

Applying the extended mapping method via Riccati equation, many exact variable separation solutions for the （2＆1 ）-dimensional variable coefficient Broer-Kaup equation are obtained. Introducing multiple valued function and Jacobi elliptic function in the seed solution, special types of periodic semifolded solitary waves are derived. In the long wave limit these periodic semifolded solitary wave excitations may degenerate into single semifolded localized soliton structures. The interactions of the periodic semifolded solitary waves and their degenerated single semifolded soliton structures are investigated graphically and found to be completely elastic.     

Doubly Periodic Propagating Wave Patterns of （2＋1）-Dimensional Maccari System

Using the variable separation approach, we obtain a general exact solution with arbitrary variable separation functions for the （2＋ 1）-dimensional Maccari system. By introducing Jacobi elliptic functions dn and nd in the seed solution, two types of doubly periodic propagating wave patterns are derived. We invest/gate the wave patterns evolution along with the modulus k increasing, many important and interesting properties are revealed.     

Multi-linear variable separation approach to nonlinear systems

The multi-linear variable separation approach is reviewed in this article. The method has been recently established and successfully solved a large number of nonlinear systems. One of the most exciting findings is that the basic multi-linear variable separation solution can be expressed by a universal formula including two (1+1)-dimensional functions, and at least one is arbitrary for integrable systems. Furthermore, the method has been extended in two different ways so as to enroll more low dimensional functions in the solution.      

Doubly Periodic Propagating Wave for （2d-1）-Dimensional Breaking Soliton Equation

Using the variable separation approach, we obtain a general exact solution with arbitrary variable separation functions for the （2＋1）-dimensional breaking soliton system. By introducing Jacobi elliptic functions in the seed solution, two families of doubly periodic propagating wave patterns are derived. We investigate these periodic wave solutions with different modulus m selections, many important and interesting properties are revealed. The interaction of Jabcobi elliptic function waves are graphically considered and found to be nonelastic.     

(2+1)维非线性Burgers方程变量分离解和新型孤波结构

利用变量分离方法，获得了(2+1)维非线性Burgers方程的变量分离解.由于在Bcklund变换和变量分离步骤中引入了作为种子解的任意函数, 因而精确解中含有三个任意函数(其中一个为条件函数),适当地选择任意函数，可以获得多种形状的扭状孤波解、周期性孤子解和格子型孤波解. 关键词： 变量分离解 非线性波方程 （2+1）维     

Maccari系统的椭圆函数传播波

基于多线性分离变量法所得(2＋1)维Maccari非线性系统的精确解，在分离变量函数中引入雅克比椭圆函数，获得两类双周期传播波模式. 椭圆函数波在不同模量取值下，具有不同的性状特点，特别是在模量极限情形下，可以约化为dromion和peakon激发形态.利用图示法对椭圆函数波的相互作用进行了探讨，发现其相互作用是非弹性的. 关键词： Maccari系统 多线性分离变量法 雅克比椭圆函数 周期波     

(2+1)维修正Veselov-Novikov系统的新型折叠子及其弹性碰撞

借助Mathematica软件，在Bcklund变换的基础上采用多线性变量分离(MLVS)方法，得到了(2+1)维修正Veselov-Novikov系统的一个含低维任意函数的新的精确解.选取合适的多值函数，构造出新型的折叠子，对其进行了分类并且研究了各种类型的二折叠子之间的完全弹性碰撞.另外还给出了折叠子与隐形折叠子的相互作用.最后把MLVS方法推广到一个新的(1+1)维非线性系统. 关键词： 修正Veselov-Novikov系统 折叠子 弹性碰撞 变量分离     

Quasi-Periodic Structures Based on Symmetrical Lucas Function of(2+1)-Dimensional Modified Dispersive Water-Wave System

By introducing the Lucas--Riccati method and a linear variable separationmethod, new variable separation solutions with arbitrary functions arederived for a (2+1)-dimensional modified dispersive water-wave system. Themain idea of this method is to express the solutions of this system aspolynomials in the solution of the Riccati equation that the symmetricalLucas functions satisfy. From the variable separation solution and byselecting appropriate functions, some novel Jacobian elliptic wave structurewith variable modulus and their interactions with dromions and peakons are investigated.