A simple but efficient approach for studying on nonlinear differential-difference equations

In this paper, we present a new approach for constructing exact solutions to nonlinear differential-difference equations (NLDDEs). By applying the new method, we have studied the saturable discrete nonlinear Schrodinger equation (SDNLSE) and obtained a number of new exact localized solutions, including discrete bright soliton solution, dark soliton solution, bright and dark soliton solution, alternating phase bright soliton solution, alternating phase dark soliton solution and alternating phase bright and dark soliton solution, provided that a special relation is bound on the coefficients of the equation among the solutions obtained.     

Camassa-Holm方程凹凸尖峰及光滑孤立子解

研究一类完全可积的新型浅水波方程Camassa-Holm方程的行波孤立子解及双孤立子解.引入凹凸尖峰孤立子及光滑孤立子的概念,研究得到该方程的行波解中具有尖峰性质的凹凸尖峰孤立子解及光滑孤立子解.同时利用Backlund变换给出该类方程的新的双孤立子解.     

Optical solitons in a power law media with fourth order dispersion

In this paper, a closed form optical soliton solution is obtained for the nonlinear Schrödinger’s equation with fourth order dispersion in a power law media. The solitary wave ansatze is used to carry out the integration of this equation. Finally, a numerical simulation is given for the closed form soliton solution.     

Sine-Gordon方程孤立子解的表达式

本文研究Sine-Gordon方程
u_xt=sinu(A)
的反散射解.给出了(A)的孤立子解的简洁表达式,并讨论了单孤立子解和双孤立子解.     

Solving soliton equations with self-consistent sources by method of variation of parameters

Starting from the solutions of soliton equations and corresponding eigenfunctions obtained by Darboux transformation, we present a new method to solve soliton equations with self-consistent sources (SESCS) based on method of variation of parameters. The KdV equation with self-consistent sources (KdVSCS) is used as a model to illustrate this new method. In addition, we apply this method to construct some new solutions of the derivative nonlinear Schrödinger equation with self-consistent sources (DNLSSCS) such as phase solution, dark soliton solution, bright soliton solution and breather-type solution.     

Weakly damped KdV soliton dynamics with the random force

The soliton dynamics in the random field is studied in the framework of the Korteweg–de Vries–Burgers equation. Asymptotic solution of this equation with weak dissipation is found and the average wave field is analyzed. All formulas can be given explicitly for the uniform (table-top) distribution function of the random field. Weakly damped KdV soliton on large times transforms to the “thick” soliton or KdV-like soliton depending from the statistical properties of the force. New scenario of KdV soliton transformation into the thick soliton and then again in KdV-like soliton is predicted for certain conditions.     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

New travelling wave solutions for a nonlinearly dispersive wave equation of Camassa-Holm equation type

In this paper, the integral bifurcation method is used to study a nonlinearly dispersive wave equation of Camassa-Holm equation type. Loop soliton solution and periodic loop soliton solution, solitary wave solution and solitary cusp wave solution, smooth periodic wave solution and non-smooth periodic wave solution of this equation are obtained, their dynamic characters are discussed. Some solutions have an interesting phenomenon that one solution admits multi-waves when parameters vary.     

导数Manakov方程的Darboux变换及其精确解

本文给出了导数Manakov方程新的Darboux变换.利用此Darboux变换得到了导数Manakov方程的精确解.最后,通过选择适当的参数,作出了孤子解的图形.     

Soliton solution and bifurcation analysis of the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation with power law nonlinearity

This paper addresses the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation with power law nonlinearity. First the soliton solution is obtained by the aid of traveling wave hypothesis and along with it the constraint conditions fall out naturally, in order for the soliton solution to exist. Subsequently, the bifurcation analysis of this equation is carried out and the fixed points are obtained. The phase portraits are also analyzed for the existence of other solutions.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics for Supersonic Soliton Propagation in the Toda Lattice Equation

We study the problem of the adjustment of an initial condition to an exact supersonic soliton solution of the Toda latice equation. Also, we study the problem of soliton propagation in the Toda lattice with slowly varying mass impurities. In both cases we obtain the full numerical solution of the soliton evolution and we develop a modulation theory based on the averaged Lagrangian of the discrete Toda equation. Unlike previous problems with coherent subsonic solutions we need to modify the averaged Lagrangian to obtain the coupling between the supersonic soliton and the subsonic linear radiation. We show how this modified modulation theory explains qualitatively in simple terms the evolution of a supersonic soliton in the presence of impurities. The quantitative agreement between the modulation solution and the numerical result is good.     

The Darboux transformation associated with two-parameter lattice soliton equation

A new discrete matrix spectral problem with two arbitrary constants is introduced. The corresponding two-parameter integrable lattice soliton equation is obtained through the discrete zero curvature representation, and the resulting integrable lattice equation reduce to the Toda lattice in rational form for a special choice of the parameters. A Darboux transformation (DT) for the lattice soliton equation is constructed. As an application, an explicit solution of the two-parameter lattice soliton equation is presented.     

New Solutions to the Ultradiscrete Soliton Equations

New solutions to the ultradiscrete soliton equations, such as the Box–Ball system, the Toda equation, etc. are obtained. One of the new solutions which we call a "negative-soliton" satisfies the ultradiscrete KdV equation (Box–Ball system) but there is not a corresponding traveling wave solution for the discrete KdV equation. The other one which we call a "static-soliton" satisfies the ultradiscrete Toda equation but there is not a corresponding traveling wave solution for the discrete Toda equation. A collision of a soliton with a negative-soliton generates many balls in a box over the capacity of the box in the Box–Ball system, while a collision of a soliton with the static-soliton describes, in the ultradiscrete limit, transmission of a soliton through junctions of a "nonuniform Toda equation." We have obtained exact solutions describing these phenomena.     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

New periodic wave and soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation

The periodic wave solutions for Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.     

The families of explicit solutions for the Hirota equation

The well‐known shallow wave equation can be reduced to the Hirota equation with the aid of corresponding transformation. We discuss its explicit solutions, including dark soliton solution, multiple soliton solution, multiple singular solution, and periodic solutions.     

Solitary wave solution for the generalized Kawahara equation

The travelling wave ansatz is used to find the solitary wave solution of the generalized Kawahara equation. The ansatz is obtained from the structure of the soliton solution of the Kawahara equation and the modified Kawahara equation. The first two integrals of motion of the generalized Kawahara equation are also computed in this work.     

Stable transmission of solitons in the complex cubic–quintic Ginzburg–Landau equation with nonlinear gain and higher-order effects

In this paper, a complex cubic–quintic Ginzburg–Landau equation (CCQGLE) is investigated. Using the asymmetric method, the analytic one-soliton solution of the CCQGLE is obtained for the first time. Through analyzing the solutions obtained, the transmission of the soliton is controlled by changing the values of related parameters. Results of this paper contribute to obtain the analytic soliton solution of the higher-order CCQGLE.