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.     

Variable Separation Solutions for the （2＋1）—Dimensional Burgers Equation

Considering that the multi-linear variable separation approach has been proved to be very useful to solve many (2 1)- dimensional intergrable systems,we obtain the variable separation solutions of the Burgers eqation with arbitrary number of variable separated functions.The Y-shaped soliton fusion phenomenon is revealed.     

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.     

Variable Separation Approach to Solve Nonlinear Systems

The variable separation approach method is very useful to solving (2 1)-dimensional integrable systems.But the (1 1)-dimensional and (3 1)-dimensional nonlinear systems are considered very little. In this letter, we extend this method to (1 1) dimensions by taking the Redekopp system as a simp!e example and (3 1)-dimensional Burgers system. The exact solutions are much general because they include some arbitrary functions and the form of the (3 1)-dimensional universal formula obtained from many (2 1)-dimensional systems is extended.     

SYMMETRIES AND DROMION SOLUTION OF A (2+1)-DIMENSIONAL NONLINEAR SOHR?DINGER EQUATION

The Painlevé property, infinitely many symmetries and exact solutions of a (2+1)-dimensional nonlinear Schr?dinger equation, which are obtained from the constraints of the Kadomtsev-Petviashvili equation, are studied in this paper. The Painlevé property is proved by the Weiss-Kruskal approach, the infinitely many symmetries are obtained by the formal series symmetry method and the dromion-like solution which is localized exponentially in all directions is obtained by a variable separation method.     

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.     

Variable Separation Solutions in （1＋1）-Dimensional and （3＋1）-Dimensional Systems via Entangled Mapping Approach

In this paper, the entangled mapping approach （EMA） is applied to obtain variable separation solutions of （1＋1）-dimensional and （3＋1）-dimensional systems. By analysis, we firstly find that there also exists a common formula to describe suitable physical fields or potentials for these （1＋1）-dimensional models such as coupled integrable dispersionless （CID） and shallow water wave equations. Moreover, we find that the variable separation solution of the （3＋1）-dimensional Burgers system satisfies the completely same form as the universal quantity U1 in （2＋1）-dimensional systems. The only difference is that the function q is a solution of a constraint equation and p is an arbitrary function of three independent variables.     

Folded Localized Excitations in a Generalized （2 ＋ 1）-Dimensional Perturbed Nonlinear Schroedinger System

Starting from a special Baecklund transform and a variable separation approach, a quite general variable separation solution of the generalized ( 2 1 )-dimensional perturbed nonlinear Schroedinger system is obtained. In addition to the single-valued localized coherent soliron excitations like dromions, breathers, instantons, peakons, and previously revealed chaotic localized solution, a new type of multi-valued (folded) localized excitation is derived by introducing some appropriate lower-dimensional multiple valued functions.     

Painlev Property,Bcklund 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 Painlev′e property of the(3+1)-dimensional Burgers equation, and then B¨acklund transformation is derived according to the truncated expansion of the obtained Painlev′e analysis. Using the B¨acklund transformation, we find the rouge wave solutions to the equation via the multilinear variable separation approach. And we also give an exact solution obtained by general variable separation approach, which is proved to possess abundant structures.     

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.     

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.     

Multi-valued solitary waves in multidimensional soliton systems

Considering that folded phenomena are rather universal in nature and some arbitrary functions can be included in the exact excitations of many (2+1)-dimensional soliton systems, we use adequate multivalued functions to construct folded solitary structures in multi-dimensions. Based on some interesting variable separation results in the literature, a common formula with arbitrary functions has been derived for suitable physical quantities of some significant (2+1)-dimensional soliton systems like the generalized Ablowitz－Kaup－Newell－Segur (GAKNS) model, the generalized Nizhnik－Novikov－Veselov (GNNV) system and the new (2+1)-dimensional long dispersive wave (NLDW) system. Then a new special type of two-dimensional solitary wave structure, i.e. the folded solitary wave and foldon, is obtained. The novel structure exhibits interesting features not found in the single valued solitary excitations.     

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.     

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.     

New Family of Exact Solutions and Chaotic Soltions of Generalized Breor-Kaup System in （2＋1）-Dimensions via an Extended Mapping Approach

Starting from an extended mapping approach, a new type of variable separation solution with arbitrary functions of generalized （2＋1）-dimensional Broer-Kaup system （GBK） system is derived. Then based on the derived solitary wave solution, we obtain some specific chaotic solitons to the （2＋1）-dimensional GBK system.     

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.     

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.     

Chaos and Fractals in a （2＋1）—Dimensional Soliton System

Considering that there are abundant coherent solitent soliton excitations in high dimensions,we reveal a novel phenomenon that the localized excitations possess chaotic and fractal behaviour in some(2 1)-dimensional soliton systems.To clarify the interesting phenomenon,we take the generalized(2 1)-dimensional Nizhnik-Novikov-Vesselov system as a concrete example,A quite general variable separation solutions of this system is derived via a variable separation approach first.then some new excitations like chaos and fractals are derived by introducing some types of lower-dimensional chaotic and fractal patterns.     

Variable separation solutions and new solitary wave structures to the （l＋l）-dimensional Ito system

A variable separation approach is proposed and extended to the (1+1)-dimensional physics system. The variable separation solution of (1+1)-dimensional Ito system is obtained. Some special types of solutions such as non-propagating solitary wave solution, propagating solitary wave solution and looped soliton solution are found by selecting the arbitrary function appropriately.     

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.