Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travellingwave form satisfies some special conditions.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

Abstract In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f （ξ）, is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

Some New Solitary Wave Solutions to a Compound KdV-Burgers Equation

In terms of the solutions of an auxiliary ordinary differential equation, a new algebraic method, which contains the terms of first-order derivative of functions f(ξ), is constructed to explore the new solitary wave solutions for nonlinear evolution equations. The method is applied to a compound KdV-Burgers equation, and abundant new solitary wave solutions are obtained. The algorithm is also applicable to a large variety of nonlinear evolution equations.     

含高阶非线性效应的薛定谔方程的精确解研究

利用孤子理论,研究了含三次和五次非线性项的非线性薛定谔方程,在参数取不同值时得到了方程的新型亮孤子解、新型暗孤子解和新的三角函数周期解。     

Explicit Exact Solutions for Some Nonlinear Partial Differential Equations with Nonlinear Terms of Any Order

In this paper, by introducing some appropriate transformation and with the help of symbolic computation, we study exact travelling wave solutions for the high-order modified Boussinesq equation, a single nonlinear reaction-diffusion equation and a generalized nonlinear Schrödinger equation with nonlinear terms of any order by use of the extended-tanh method. Thus, some new exact travelling-wave solutions, which contain kink-shaped solitons, bell-shaped solitons, periodic solutions, combined formal solitons, rational solutions and singular solitons for these equations, are obtained.     

New Solitary Wave Solutions to the KdV-Burgers Equation

Based on the analysis on the features of the Burgers equation and KdV equation as well as KdV-Burgers equation, a superposition method is proposed to construct the solitary wave solutions of the KdV-Burgers equation from those of the Burgers equation and KdV equation, and then by using it we obtain many solitary wave solutions to the KdV-Burgers equation, some of which are new ones.PACS: 02.30.Jr; 03.65.Ge     

Solitary Wave Solutions of a Generalized Derivative Nonlinear Schrodinger Equation

With the aid of a class of nonlinear ordinary differential equation （ODE） and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Sehrodinger equation （GDNLSE） have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.     

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.     

Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation

We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.     

Solitary Wave Solutions of a Generalized Derivative Nonlinear Schrödinger Equation

With the aid of a class of nonlinear ordinary differential equation (ODE) and its various positive solutions, four types of exact solutions of the generalized derivative nonlinear Schrödinger equation (GDNLSE) have been found out, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal traveling wave solution, provided that the coefficients of GDNLSE satisfy certain constraint conditions. For more general GDNLSE, the similar results are also given.     

New Travelling Wave Solutions to Compound KdV-Burgers Equation

The compound KdV-Burgers equation and combined KdV-mKdV equation are real physical models concerning many branches in physics. In this paper, applying the improved trigonometric function method to these equations, rich explicit and exact travelling wave solutions, which contain solitary-wave solutions, periodic solutions, and combined formal solitary-wave solutions, are obtained.     

Jacobi Elliptic Function Solutions of Some Nonlinear Evolution Equations

In this letter, the modified Jacobi elliptic function expansion method is extended to solve M-coupled KdV equation, M-coupled Ito equation, vKdV equation, and AKNS equation. Some new Jacobi elliptic function solutions are obtained by using Mathematica. When the modulus m → 1, those periodic solutions degenerate as the corresponding soliton solutions.     

Jacobi Elliptic Function Solutions of Some Nonlinear Evolution Equations

In this letter, the modified Jacob/elliptic function expansion method is extended to solve M-coupled KdV equation, M-coupled Ito equation, vKdV equation, and AKNS equation. Some new Jacob/elliptic function solutions are obtained by using Mathematica. When the modulus m→1, those periodic solutions degenerate as the corresponding soliton solutions.     

Exact Analytical Solutions of the Generalized Calogero-Bogoyavlenskii-Schiff Equation Using Symbolic Computation

By means of generalized Riccati equation expansion method and symbolic computation, some exact analytical solutions, which contain soliton-like solutions and periodic-like solutions to the generalized Calogero-Bogoyavlenskii-Schiff (GCBS) equation, are obtained. From our results, the solitary-wave solutions and previously known soliton-like solutions of the special cases of GCBS equation can be recovered.     

Recursion Formulas to Find Solutions of Some Nonlinear Evolution Equations

New recursion formulas are obtained for finding solutions of some nonlinear evolution equations.     

New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schrödinger Equation

Some new exact travelling wave and period solutions of discrete nonlinear Schrödinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differential-different models.     

New Exact Traveling Wave Solutions for Compound KdV-Type Equation with Nonlinear Terms of Any Order

The compound KdV-type equation with nonlinear terms of any order is reduced to the integral form. Using the complete discrimination system for polynomial, its all possible exact traveling wave solutions are obtained. Among those, a lot of solutions are new.