Multilinear Variable Separation Approach in （3＋1）-Dimensions： the Burgers Equation

The multi-linear variable separation approach has been proved to be very useful in solving many (2 1)-dimensional integrable systems. Taking the (3 l)-dimeusional Burgers equation as a simple example, here we extend the multi-linear variable separation approach to (3 l )-dimensions. The form of the universal formula obtained from many (2 l )-dimensional system is still valid. However, a more general arbitrary function (with three independentvariables) has been included in the formula. Starting from the universal formula, one may obtain abundant (3 l )-dimensional localized excitations. In particular, we display a special paraboloid-type camber soliton solution and a dipole-type dromion solution which is localized in all directions.     

A Series of Variable Separation Solutions and New Soliton Structures of （2＋1）-Dimensional Korteweg-de Vries Equation

Variable separation approach is introduced to solve the （2＋1）-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model （24）. Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.     

Multi-linear Variable Separation Approach to Solve a （1＋1）-Dimensional Coupled Integrable Dispersionless System

The multi-linear variable separation approach （MLVSA ） is very useful to solve （2＋1）-dimensional integrable systems. In this letter, we extend this method to solve a （1＋1）-dimensional coupled integrable dispersion-less system. Namely, by using a Backlund transformation and the MLVSA, we find a new general solution and define a new ＂universal formula＂. Then, some new （1＋1）-dimensional coherent structures of this universal formula can be found by selecting corresponding functions appropriately. Specially, in some conditions, bell soliton and kink soliton can transform each other, which are illustrated graphically.     

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.     

New Exact Solution to （3＋1）-Dimensional Burgers Equation

In this Letter, using Ba^ecklund transformation and the new variable separation approach, we find a new general solution to the (3 1)-dimensional Burgers equation. The form of the universal formula obtained from many (2 1)-dimensional systems is extended. Abundant localized coherent structures can be found by seclecting corresponding functions appropriately.     

A Note on ＂Doubly Periodic Propagating Wave for （2＋1）-Dimensional Breaking Soliton Equation＂

In a recent paper [Commun. Theor. Phys. （Beijing, China） 49 （2008） 268], Huang et al. gave a general variable separation solution to the （2＋1）-dimensional breaking soliton equation via a special Biicldund transformation and the variable separation approach. In terms of the derived variable separation solution and by introducing Jacobi elliptic functions, they claimed that nonelastic types of interaction between Jacobi elliptic function waves are investigated both analytically and graphically. We show that some inappropriateness or errors exist in their paper, and say nothing of the conclusion that some nonelastic types of interaction between Jacobi elliptic function waves in the （2＋1）-dimensional breaking soliton equation have been found.     

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.     

Painleve Property,Beicklund Transformations and Rouge Wave Solutions of （3 ＋ 1）-Dimensional Burgers Equation

Burgers equation is the simplest one in soliton theory, which has been widely applied in almost all the physical branches. In this paper, we discuss the Painleve property of the （3＋1）-dimensional Burgers equation, and then Becklund transformation is derived according to the truncated expansion of the obtained Painleve analysis. Using the Backlund transformation, we find the rouge wave solutions to the equation via the multilinear variable separation approach. And we aiso give an exact solution obtained by general variable separation approach, which is proved to possess abundant structures.     

New Exact Solutions and Interactions Between Two Solitary Waves for （3＋1）-Dimensional Jimbo-Miwa System

By means of an extended mapping approach and a linear variable separation approach, a new family of exact solutions of the （3＋1）-dimensional Jimbo-Miwa system is derived. Based on the derived solitary wave solution, we obtain some special localized excitations and study the interactions between two solitary waves of the system.     

Annihilation Solitons and Chaotic Solitons for the （2＋1）-Dimensional Breaking Soliton System

By means of an improved mapping method and a variable separation method, a series of variable separation solutions including solitary wave solutions, periodic wave solutions and rational function solutions） to the （2＋1）-dimensional breaking soliton system is derived. Based on the derived solitary wave excitation, we obtain some special annihilation solitons and chaotic solitons in this short note.     

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.     

Study of （2＋1）-Dimensional Higher-Order Broer-Kaup System

Painleve property and infinite symmetries of the （2＋1）-dimensional higher-order Broer-Kaup （HBK） system are studied in this paper. Using the modified direct method, we derive the theorem of general symmetry gro.ups to （2＋1）-dimensional HBK system. Based on our theorem, some new forms of solutions are obtained. We also find infinite number of conservation laws of the （2＋1）-dimensional HBK system.     

New Baecklund Transformation and New Exact Solutions to （2＋1）-Dimensional KdV Equation

A new Baecklund transformation for (2 1)-dimensional KdV equation is first obtained by using homogeneous balance method. And making use of the Baecklund transformation and choosing a special seed solution, we get special types of solitary wave solutions. Finally a general variable separation solution containing two arbitrary functions is constructed, from which abundant localized coherent structures of the equation in question can be induced.     

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.     

New Exact Solutions for （1 ＋ 1）-Dimensional Dispersion-Less System

Using improved homogeneous balance method, we obtain complex function form new exact solutions for the （1＋1）-dimensional dispersion-less system, and from the exact solutions we derive real function form solution of the field u. Based on this real function form solution, we find some new interesting coherent structures by selecting arbitrary functions appropriately.     

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Rieatti equation expansion method, we present explicit solutions of the （2＋1）-dimensional variable coefficients Broer-Kaup （VCBK） equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.     

Multi Dromion-Solitoff and Fractal Excitations for （2＋1）-Dimensional Boit i-Leon-Manna-Pempinelli System

In this short note, a new projective equation （Ф = σФ ＋ Ф^2） is used to obtain the variable separation solutions with two arbitrary functions of the （2＋1）-dimensional Boiti-Leon-Manna Pempinelli system （BLMP）. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as multi dromion-solitoffs and fractal-solitons are investigated.     

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

In this work, we reveal a novel phenomenon that the localized coherent structures of some (2 1)-dimensional physical models possess chaotic and fractal behaviors. To clarify these interesting phenomena, we take the (2 l)-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 located coherent soliton excitations like dromions, lumps, rings, peakons, and oscillating soliton excitations, some new excitations with chaotic and fractal behaviors are derived by introducing some types of lower dimensional chaotic and fractal 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.     

Modulational Instability and Variable Separation Solution for a Generalized （2＋1）-Dimensional Hirota Equation

It is demonstrated by the linear modulational instability analysis that a generalized （2＋1）-dimensional Hirota equation is modulationally stable. Then, a B？cklund transformation （BT） is obtained by means of the truncated Painlevé approach. Using the BT, the model is transformed to a system of equations, which finally leads to a special variable separation solution with arbitrary functions.