A Novel Class of Coherent Localized Structures for the Maccari System

By means of the Baecklund transformation, a quite general variable separation solution of the (2 1)-dimensional Maccari systems is derived. In addition to some types of the usual localized excitations such as dromion, lumps, ring soliton and oscillated dromion, breathers solution, fractal-dromion, fractal-lump and chaotic soliton structures can be easily constructed by selecting the arbitrary functions appropriately, a new novel class of coherent localized structures like peakon solution and compacton solution of this new system are found by selecting apfropriate functions.     

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.     

Rich Localized Coherent Structures of the (2+1)-Dimensional Broer-Kaup-Kupershmidt Equation

Using a Backlund transformation and the variable separation approach, we find there exist abundant localized coherent structures for the (2 + 1)-dimensional Broer-Kaup-Kupershmidt (BKK) system. The abundance of the localized structures for the model is introduced by the entrance of an arbitrary function of the seed solution. For some specialselections of the arbitrary function, it is shown that the localized structures of the BKK equation may be dromions, lumps, ring solitons, peakons, or fractal solitons etc.     

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.     

Fractal Dromion, Fractal Lump, and Multiple Peakon Excitations in a New (2+1)-Dimensional Long Dispersive Wave System

By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: λqt + qxx - 2q ∫ (qr)xdy ＝ 0, λrt - rxx + 2r ∫(qr)xdy ＝ 0, is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, farctal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.     

Variable Separation Approach to Solve a (2+1)-Dimensional Integrable System

In this letter, starting from a B\"{a}cklund transformation, a general solution of a (2+1)-dimensional integrable system is obtained by using the new variable separation approach.     

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     

Exact Excitation and Abundant Localized Coherent Soliton Structures of (2+1)-Dimensional Perturbed AKNS System

A simple and direct method is applied to solving the (2+1)-dimensional perturbed Ablowitz-Kaup-Newell-Segur system (PAKNS). Starting from a special Backlund transformation and the variable separation approach, we convert the PAKNS system into the simple forms, which are four variable separation equations, then obtain a quite generalsolution. Some special localized coherent structures like fractal dromions and fractal lumps of this model are constructed by selecting some types of lower-dimensional fractal patterns.     

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.     

Notes on Multi-linear Variable Separation Approach

The multi-linear variable separation approach method is very useful to solve (2+1)-dimensional integrable systems. In this letter, we extend this method to solve (1+1)-dimensional Boiti system, (2+1)-dimensional Burgers system, (2+1)-dimensional breaking soliton system, and (2+1)-dimensional Maccari system. Some new exact solutions are obtained and the universal formula obtained from many (2+1)-dimensional systems is extended or modified.     

New Exact Solutions and Localized Structures for （3＋1）-Dimensional Burgers System

With an extended mapping approach and a linear variable separation method, new families of variable separation solutions （including solitary wave solutions, periodic wave solutions, and rational function solutions） with arbitrary functions for （3＋1）-dimensionai Burgers system is derived. Based on the derived excitations, we obtain some novel localized coherent structures and study the interactions between solitons.     

Localized Structures on Periodic Background Wave of （2＋1）-Dimensional Boiti-Leon-Pempinelli System via an Object Reduction

In this paper, we present an object reduction for nonlinear partial differential equations. As a concrete example of its applications in physical problems, this method is applied to the （2＋1）-dimensional Boiti-Leon-Pempinelli system, which has the extensive physics background, and an abundance of exact solutions is derived from some reduction equations. Based on the derived solutions, the localized structures under the periodic wave background are obtained.     

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.     

Semifolded Localized Structures in Three-Dimensional Soliton Systems

By means ofa Painlev6 Backlund transformation and a multi-linear variable separation approach, abundant localized coherent excitations of the three-dimensional Broer Kaup Kupershmidt system with variable coeft~cients are derived. There are possible phase shifts for the interactions of the three-dimensional novel localized structures discussed in this paper.     

Fission and Fusion of Localized Coherent Structures for a Higher-Order Broer-Kaup System

Starting from a Bäcklund transformation and taking a special ansatz for the function f, we can obtain a much more general expression of solution that includes some variable separated functions for the higher-order Broer-Kaup system. From this expression, we investigate the interactions of localized coherent structures such as the multi-solitonic excitations and find the novel phenomenon that their interactions have non-elastic behavior because the fission/fusion may occur after the interaction of each localized coherent structure.     

Evolutional Properties of Localized Excitations for Generalized Broer-Kaup System in （2＋1） Dimensions

Using a special Painleve-Baecklund transformation as well as the extended mapping approach and the linear superposition theorem, we obtain new families of variable separation solutions to the （2＋1）-dimensional generalized Broer-Kaup （GBK） system. Based on the derived exact solution, we reveal some novel evolutional behaviors of localized excitations, i.e. fission and fusion phenomena in the （2＋1）-dimensional GBK system.     

Fission and Fusion in the New Localized Structures to the Integrable (2 + 1)-Dimensional Higher-Order Broer–Kaup System

By means of the Weiss–Tabor–Carnevale (WTC) truncation method and the general variable separation approach (GVSA), analytical investigation of the integrable (2+1)-dimensional higher-order Broer–Kaup (HBK) system shows, due to the possibility of selecting three arbitrary func.tions, the existence of interacting coherent excitations such as dromions, solitons, periodic solitons, etc. The interaction between some of the localized solutions are elastic because they pass through each other and preserve their shapes and velocities, the only change being the phase shift. However, as for some soliton models, completely non-elastic interactions have been found in this model. These non-elastic interactions are characterized by the fact that, at a specific time, one soliton may fission to two or more solitons; or on the contrary, two or more solitons will fuse to one soliton.     

New Types of Localized Coherent Structures in the Bogoyavlenskii-Schiff Equation

Based on the singular structure analysis, we derive some new types of localized coherent structures for the Bogoyavlenskii-Schiff equation by suitably utilizing the arbitrary function present in the singular manifold equations. PACS 05.45. Yv, 02.30.Ik, 02.30.Jr.