Multi-linear Variable Separation Approach to Solve a (2+1)-DimensionalGeneralization of Nonlinear Schroedinger System

By using a Baecklund transformation and the multi-linear variable separationapproach, we find a new general solution of a (2+1)-dimensional generalization of the nonlinearSchroedinger system. The new 'universal' formula is defined, and then, rich coherent structures canbe found by selecting corresponding functions appropriately.

Variable Separation Approach to Solve （2 ＋ 1）-Dimensional Generalized Burgers System： Solitary Wave and Jacobi Periodic Wave Excitations

By means of the standard truncated Painlevé expansion and a variable separation approach, a general variable separation solution of the generalized Burgers system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, peakons, foldons, and previously revealed chaotic and fractal localized solutions, some new types of excitations — compacton and Jacobi periodic wave solutions are obtained by introducing appropriate lower dimensional piecewise smooth functions and Jacobi elliptic functions.

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.

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

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

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.

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.

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-F1）-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.

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.

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

Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more generalexpression 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.

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 B?cklund 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 general solution. 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.

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: 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, fractal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.

New periodic wave solutions, localized excitations and their interaction for (2+1)-dimensional Burgers equation

Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for （2＋l）-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions （which contains two arbitrary functions）. Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave （called semi-localized） patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations （two-dromion solution）.

Rational Solutions in a Coupled Burgers System

Three types of the rational solutions for a new coupled Burgers system are studied in detail in terms of the reduction and decoupled procedures. The first two types of rational solutions are singular and valid for one type of model parameter c 〉 0, and another type of rational solutions is nonsingular at any type and valid for another type of model parameter c 〈 0.

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 partial differential equation. Applying the B?cklund transformation and introducing the arbitrary functions of the seed solutions, the abundance of the localized structures of this model are derived. Some special types of solutions solitoff, dromions, dromion lattice, breathers and instantons are discussed by selecting the arbitrary functions appropriately. The breathers may breath in their amplititudes, shapes, distances among the peaks and even the number of the peaks.

A Discrete Lax-Integrable Coupled System Related to Coupled KdV and Coupled mKdV Equations

A modified Korteweg-de Vries （mKdV） lattice is found to be also a discrete Korteweg-de Vries （KdV） equation. A discrete coupled system is derived from the single lattice equation and its Lax pair is proposed. The coupled system is shown to be related to the coupled KdV and coupled mKdV systems which are widely used in physics.

Interaction Between Line Soliton and Algebraic Soliton for Asymmetric Nizhnik-Novikov-Veselov Equation

Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for （2＋1）-dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton： 1） the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2） the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.

Standing,Periodic and Solitary Waves in (1+1)-Dimensional Caudry-Dodd-Gibbon-Sawada-Kortera System

In the paper, the variable separation approach, homoclinic test technique and bilinear method are successfullyextended to a (1+1)-dimensional Caudry-Dodd-Gibbon-Sawada-Kortera (CDGSK) system, respectively.Basedon the derived exact solutions, some significant types of localized excitations such as standing waves, periodic waves,solitary waves are simultaneously derived from the (1+1)-dimensional Caudry-Dodd-Gibbon-Sawada-Kortera systemby entrancing appropriate parameters.

Solitons with Periodic Behavior in Korteweg-de Vries Type Models Related toSchrodinger System

The linear variable separation approach is successfully extended to(1+1)-dimensional Korteweg-de Vries (KdV) type models related to Schrodinger system. Somesignificant types of solitons such as compaction, peakon, and loop solutions with periodic behaviorare simultaneously derived from the (l+l)-dimensional soliton system by entrancing appropriatepiecewise smooth functions and multivalued functions.

Soliton Fission and Fusion in (2+l)-Dimensional Boiti-Leon-Pempinelli System

By means of a special Painleve—Baecklund transformation and a multilinearvariable separation approach, an exact solution with arbitrary functions of the (2+1)-dimensionalBoiti-Leon-Pempinelli system (BLP) is derived. Based on the derived variable separation solution, weobtain some special soliton fission and fusion solutions for the higher dimensional BLP system.

Analytical Study of Nonlinear Dust Acoustic Waves in Two-Dimensional Dust Plasma with Dust Charge Variation

The nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation is analytically investigated by using the formally variable separation approach. New analytical solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma for the first time. We derive exact analytical expressions for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation.