Solitary Waves of a Perturbed sine—Gordon Equation

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves,different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained.All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.     

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota－Satsuma KdV system

In this paper,we study the generalized coupled Hirota-Satsuma KdV system by using the new generalized transformation in homogeneous balance method.As a result,many explicit exact solutions,which contain new solitary wave solutions,periodic wave solutions,and the combined formal solitary wave solutions,and periodic wave solutions ,are obtained.     

New Solitary Solutions of （2＋1）-Dimensional Variable Coefficient Nonlinear Schrodinger Equation with an External Potential

By a series of transformations, the （2＋1）-dimensional variable coefficient nonlinear Schr？dinger equation can turn to the Klein-Gordon equation. Many new double travelling wave solutions of the Klein-Gordon equation are obtained. Thus, the new solitary solutions of the variable coefficient nonlinear Schr？inger equation with an external potential can be found.     

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.     

Bound States of the S-Wave Equation with Equal Scalar and Vector Standard Eckart Potential

A supersymmetric technique for the bound-state solutions of the s-wave Klein-Gordon equation with equal scalar and vector standard Eckart-type potential is proposed. Its exact solutions are obtained. Possible generalization of our approach is outlined.     

Abundant New Explicit Exact Soliton—Like Solutions and Painleve Test for the Generalized Burgers Equation in （2＋1）—Dimensional Space

We obtain Backlund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2 1)-dimensional space,ut 1/2(uδy^-1ux)x-uxx=0,by using the extended homogeneous balance method.As is well known,the introduction of the concept of dromions (the exponentially localized solutions in (2 1)-dimensional space)has triggered renewed interest in (2 1)-dimensional soliton systems.The solutions obtained are used to show that the variable ux admits exponentially localized solutions rather than the physical field u(x,y,t) itself.In addition,it is shown that the equation passes Painleve test.     

Qualitative Analysis and Explicit Exact Solitary,Kink and Anti-Kink Wave Solutions of the Generalized Nonlinear Schrdinger Equation with Parabolic Law Nonlinearity

The generalized nonlinear Schrdinger equation with parabolic law nonlinearity is studied by using the factorization technique and the method of dynamical systems.From a dynamic point of view,the existence of smooth solitary wave,kink and anti-kink wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given.Also,all possible explicit exact parametric representations of the waves are presented.     

Searching for the （3＋1）—Dimensional Painleve Integrable Model and its Solitary Wave Solution

A （3 1)-dimensional integrable model constructed by conformal invariants is proven to be integrable.The solitary wave solution of the model is obtained by a simple algebraic transformation relation between the (3 1)-dimensional Harry-Dym equation and the cubic nonlinear Klein-Gordon equation.     

New Symmetry Reductions,Dromions—Like and Compacton Solutions for a 2D BS（m,n） Equations Hierarchy with Fully Nonlinear Dispersion

We have found two types of important exact solutions,compacton solutions,which are solitary waves with the property that after colliding with their own kind,they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction,in the (1 1)D,(1 2)D and even (1 3)D models,and dromion solutions (exponentially decaying solutions in all direction) in many (1 2)D and (1 3)D models.In this paper,symmetry reductions in (1 2)D are considered for the break soliton-type equation with fully nonlinear dispersion (called BS(m,n) equation)ut b(u^m)xxy 4b(u^n δx^-1uy)x=0,which is a generalized model of (1 2)D break soliton equation ut buxxy 4buuy 4buxδx^-1uy=0,by using the extended direct reduction method.As a result,six types of symmetry reductions are obtained.Starting from the reduction equations and some simple transformations,we obtain the solitary wavke solutions of BS(1,n) equations,compacton solutions of BS(m,m-1) equations and the compacton-like solution of the potential form (called PBS(3,2)) ωxt b(ux^m)xxy 4b(ωx^nωy)x=0.In addition,we show that the variable ∫^x uy dx admits dromion solutions rather than the field u itself in BS(1,n) equation.     

Lie group analysis,numerical and non-traveling wave solutions for the （2＋l）-dimensional diffusion-advection equation with variable coefficients

In this paper, the variable-coefficient diffusion-advection （DA） equation, which arises in modeling various physical phenomena, is studied by the Lie symmetry approach. The similarity reductions are derived by determining the complete sets of point symmetries of this equation, and then exact and numerical solutions are reported for the reduced second-order nonlinear ordinary differential equations. Further, an extended （Gl/G）-expansion method is applied to the DA equation to construct some new non-traveling wave solutions.