Fractal Structures and Chaotic Behaviors in a （2＋1）-Dimensional Nonlinear System

With a new projective equation, a series of solutions of the （2-J-1）-dimensional dispersive long-water wave system （LWW） is derived. Based on the derived solitary wave solution, we obtain some special fractal localized structures and chaotic patterns.     

Complex Wave Solutions for （2＋1）-Dimensional Modified Dispersive Water Wave System

The extended Riccati mapping approach^[1] is further improved by generalized Riccati equation, and combine it with variable separation method, abundant new exact complex solutions for the （2＋1）-dimensional modified dispersive water-wave （MDWW） system are obtained. Based on a derived periodic solitary wave solution and a rational solution, we study a type of phenomenon of complex wave.     

New Coherent Structures in the Generalized（2＋1）—Dimensional Nizhnik—Novikov—Veselov System

In high dimensions there are abundant coherent soliton excitations.From the known variable separation solutions for the generalized(2 1)-dimensional Nizhnik-Novikov-Veselov system.two kinds of new coherent structures in this system are obtained.Some interesting novel features of these structures are revealed.     

Some New Exact Solutions to the Dispersive Long－Wave Equation in（2＋1）－Dimensional Spaces

In this paper, by using a further extended tanh method- and symbolic computation system, some new soliton-like and period form solutions of the dispersive long-wave equation in (2 l )-dimensional spaces are obtained.     

Exotic Localized Coherent Structures of the （2＋1）—Dimensional Dispersive Long—Wave Equation

This article is concerned with the extended homogeneous balance method for studying the abundant localized solution structures in the (2 1)-dimensional dispersive long-wave equations uty ηxx (u^2)xy/2=0,ηt (uη u uxy)x=0.Starting from the homogeneous balance method,we find that the richness of the localized coberent structures of the model is caused by the entrance of two variable-separated arbitrary functions.for some special selections of the arbitrary functions,it is shown that the localized structures of the model may be dromions,lumps,breathers,instantons and ring solitons.     

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.     

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 separation method, new variable separation solutions with arbitrary functions are derived for a （2＋1）-dimensional modified dispersive water-wave system. The main idea of this method is to express the solutions of this system as polynomials in the solution of the Riecati equation that the symmetrical Lucas functions satisfy. From the variable separation sohition and by selecting appropriate functions, some novel Jacobian elliptic wave structure with variable modulus and their interactions with dromions and peakons are investigated.     

sl玻色子体系在U（2l＋1）←→0（2l＋2）过渡区的严格解

利用无穷维李代数方法得到了相互作用sl玻色子体系在U（2l 1)←→0(2l 2)过渡区的能谱和波函数的严格解。给出了该系统Bethe假定方程的数值解法。     

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.     

Localized Coherent Structures with Chaotic and Fractal Behaviors in a(2+1)-Dimensional Modified Dispersive Water-Wave System

In this work, we reveal a novel phenomenon that the localized coherent structures of some (2 1 )-dimensionalphysical models possess chaotic and fractal behaviors. To clarify these interesting phenomena, we take the (2 1)-dimensional modified dispersive water-wave system as a concrete example. Starting from a variable separation approach,a general variable separation solution of this system is derived. Besides the stable localized coherent soliton excitationslike dromions, lumps, rings, peakons, and oscillating soliton excitations, some new excitations with chaotic and fractalbehaviors are derived by introducing some types of lower dimensional chaotic and fractal patterns.     

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.     

Symmetry Reduction of the (2+1)-Dimensional Modified Dispersive Water-Wave System

Using the standard truncated Painlevé expansion, the residual symmetry of the (2+1)-dimensional modified dispersive water-wave system is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived by utilizing the obtained symmetries. The symmetries of the system are also derived through the Clarkson-Kruskal direct method, and several types of explicit reduction solutions relate to the trigonometric or the hyperbolic functions are obtained. Finally, some special solitons are depicted from one of the solutions.     

New Exact Solutions and Evolution of Solitons in Background Waves for (2+1)-Dimensional Modified Dispersive Water-Wave System

With the help of the conditional similarity reduction method, some new exact solutions of the (2+1)-dimensional modified dispersive water-wave system (MDWW) are obtained. Based on the derived solution, we investigate the evolution of solitons in the background waves.     

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.     

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.     

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.     

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.     

General Solution and Localized Coherent Soliton Structures of the (2+1)-Dimensional Generalized Davey-Stewarson Equations

In this paper, the variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized Davey-Stewarson equations: iqt 1/2(qxx qyy) (R S)q = O, Rx=-σ/2|q|2y Sy = -σ/2|q|2/x.Applying a special Backlund transformation and introducing arbitrary functions of the seed solutions, an abundance of the localized structures of this model is derived. By selecting the arbitrary functions appropriately, some special typesof localized excitations such as dromions, dromion lattice, breathers, and instantons are constructed.