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.     

Soliton molecules and mixed solutions of the(2+1)-dimensional bidirectional Sawada-Kotera equation

In this paper, we investigate some new interesting solution structures of the(2+1)-dimensional bidirectional Sawada–Kotera(bSK) equation. We obtain soliton molecules by introducing velocity resonance. On the basis of soliton molecules, asymmetric solitons are obtained by changing the distance between two solitons of molecules. Based on the N-soliton solutions,several novel types of mixed solutions are generated, which include the mixed breather-soliton molecule solution by the module resonance of the wave number and partial velocity resonance,the mixed lump-soliton molecule solution obtained by partial velocity resonance and partial long wave limits, and the mixed solutions composed of soliton molecules(asymmetric solitons), lump waves, and breather waves. Some plots are presented to clearly illustrate the dynamic features of these solutions.     

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.     

High-Dimensional Integrable Models with Conformal Invariance

Using the (2 1)-dimensional Schwartz dcrivative, the usual (2 1)-dimensional Schwartz Kadomtsev-Petviashvili (KP) equation is extended to (n 1)-dimensional conformal invariance equation. The extension possesses Painlcvc property. Some (3 1)-dimensional examples are given and some single three-dimensional camber soliton and two spatial-plane solitons solutions of a (3 1)-dimensional equation are obtained.     

LoLocalized Spatial Soliton Excitations in(2+1)-Dimensional Nonlinear Schrdinger Equation with Variable Nonlinearity and an External Potential

We report on the localized spatial soliton excitations in the multidimensional nonlinear Schrdinger equation with radially variable nonlinearity coefficient and an external potential.By using Hirota's binary differential operators,we determine a variety of external potentials and nonlinearity coefficients that can support nonlinear localized solutions of different but desired forms.For some specific external potentials and nonlinearity coefficients,we discuss features of the corresponding(2+1)-dimensional multisolitonic solutions,including ring solitons,lump solitons,and soliton clusters.     

Fission and Fusion of Solitons for the （1＋1）—Dimensional Kupershmidt Equation

By means of the heat conduction equation and the standard truncated Painleve expansion,the (1 1)-dimensional Kupershmidt equation is solved.Some significant exact multi-soliton solutions are given.Especially,for the interaction of the multi-solitons of the Kupershmidt equation,we find that a single(resonant)kink or bell soliton may be fissioned to several kink or bell solitons,Inversely,several kink or bell solitons may also be fused to one kink or bell soliton.     

Exact Soliton Solutions for the (2+1)-Dimensional Coupled Higher-Order Nonlinear Schr¨odinger Equations in Birefringent Optical-Fiber Communication

In birefringent optical fibers, the propagation of femtosecond soliton pulses is described by coupled higherorder nonlinear Schrdinger equations. In this paper, we will investigate the bright and dark soliton solutions of(2+1)-dimensional coupled higher-order nonlinear Schrdinger equations, with the aid of symbolic computation and the Hirota method. On the basis of soliton solutions, we test and discuss the interactions graphically between the solitons in the x-z, x-t, and z-t planes.     

Resonance Y-type soliton solutions and some new types of hybrid solutions in the(2+1)-dimensional Sawada–Kotera equation

In this paper, based on N-soliton solutions, we introduce a new constraint among parameters to find the resonance Y-type soliton solutions in (2+1)-dimensional integrable systems. Then, we take the (2+1)-dimensional Sawada–Kotera equation as an example to illustrate how to generate these resonance Y-type soliton solutions with this new constraint. Next, by the long wave limit method, velocity resonance and module resonance, we can obtain some new types of hybrid solutions of resonance Y-type solitons with line waves, breather waves, high-order lump waves respectively. Finally, we also study the dynamics of these interaction solutions and indicate mathematically that these interactions are elastic.     

Nonautonomous Solitons in the(3+1)-Dimensional Inhomogeneous Cubic-Quintic Nonlinear Medium

We have constructed explicit nonautonomous soliton solutions of the generalized nonlinear Schrdinger equation in the(3+1)-dimensional inhomogeneous cubic-quintic nonlinear medium.The gain parameter has no effects on the motion of the soliton's phase or their velocities,and it affects just the evolution of their peaks.As two examples,we discuss the propagation of nonautonomous solitons in the periodic distributed amplification system and the exponential dispersion decreasing system.Results show that the presence of the chirp not only makes the intensity of solitons weaken more promptly,but also broadens their width.     

Symmetry Reduction and Exact Solutions of the （3＋1）-Dimensional Breaking Soliton Equation

By means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the （3＋1）-dimensional breaking soliton equation. Based on the relationship, a new solution is obtained by using a given solution of the equation. The symmetry is also obtained for the （3＋1）-dimensional breaking soliton equation. By using the equivalent vector of the symmetry, we construct a seven-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, the （3＋1）-dimensional breaking soliton equation is reduced and some solutions to the reduced equations are obtained. Furthermore, some new explicit solutions are found for the （3＋ 1）-dimensional breaking soliton equation.     

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.     

A new (2+1)-dimensional supersymmetric Boussinesq equation and its Lie symmetry study

According to the conjecture based on some known facts of integrable models, a new (2+1)-dimensional supersymmetric integrable bilinear system is proposed. The model is not only the extension of the known (2+1)-dimensional negative Kadomtsev--Petviashvili equation but also the extension of the known (1+1)-dimensional supersymmetric Boussinesq equation. The infinite dimensional Kac--Moody--Virasoro symmetries and the related symmetry reductions of the model are obtained. Furthermore, the traveling wave solutions including soliton solutions are explicitly presented.     

Instantaneous solitons and fractal solitons for a (2+1)-dimensional nonlinear system

By an improved projective equation approach and a linear variable separation approach, a new family of exact solutions of the (2+1)-dimensional Broek--Kaup system is derived. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as instantaneous solitons and fractal solitons are investigated.     

Backlund transformation and multiple soliton solutions for the （3＋1）—dimensional Jimbo—Miwa equation

We study an approach to constructing multiple soliton solutions of the (3 1)-dimensional nonlinear evolution equation.We take the (3 1)-dimensional Jimbo-Miwa(JM) equation as an example.Using the extended homogeneous balance method,one can find a backlund transformation to decompose the (3 1)-dimensional JM equation into a linear partial differential equation and two bilinear partial differential equations.Starting from these linear and bilinear partial differential equations,some multiple soliton solutions for the (3 1)-dimensional JM equation are obtained by introducing a class of formal solutions.     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion(CTE) method is applied to the(2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution,and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlev′e truncated expansion method.And we investigate interactive properties of solitons and periodic waves.     

The Soliton Solution of （3＋1）—Dimensional Kadomtsev—Petviashvili Equations

A simple algebraic transformation relation of a special type of solution between the (3 1)-dimensional Kadomtsev-petviashvili(KP) equation and the cubic nonlinear Klein-Gordon equation (NKG) is established.Using known solutions of the NKG equation,we can obtain many soliton solutions and periodic solution of the (3 1)-dimensional KP equation.     

General Solution and Fractal Localized Structures for the （2＋1）—Dimensional Generalized Ablowitz—Kaup—Newell—Segur System

Using the standard truncated Painleve expansions,we derive a quite general solution of the (2 1)-dimensional generalized Ablowitz-Kaup-Newell-Segur system.Except for the usual localized solutions,such as dromions,lumps,ring soliton solutions,etc,some special localized excitations with fractal behaviour i.e.the fractal dromion and fractal lump excitations,are obtained by some types of lower-dimensional 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.     

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.     

Objective Reduction Solutions to Higher-Order Boussinesq System in （2＋1）-Dimensions

With the help of an objective reduction approach （ORA）, abundant exact solutions of （2＋1）-dimensional higher-order Boussinesq system （including some hyperboloid function solutions, trigonometric function solutions, and a rational function solution） are obtained. It is shown that some novel soliton structures, like single linearity soliton structure, breath soliton structure, single linearity y-periodic solitary wave structure, libration dromion structure, and kink-like multisoliton structure with actual physical meaning exist in the （2＋1）-dimensional higher-order Boussinesq system.