The superposition method in seeking the solitary wave solutions to the KdV-Burgers equation

In this paper, starting from the careful analysis on the characteristics of the Burgers equation and the KdV equation as well as the KdV-Burgers equation, the superposition method is put forward for constructing the solitary wave solutions of the KdV-Burgers equation from those of the Burgers equation and the KdV equation. The solitary wave solutions for the KdV-Burgers equation are presented successfully by means of this method.     

NEW EXPLICIT AND EXACT TRAVELLING WAVE SOLUTIONS FOR A COMPOUND KdV－BURGERS EQUATION

In this paper, new explicit and exact travelling wave solutions for a compound KdV－Burgers equation are obtained by using the hyperbola function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV－Burgers and mKdV equations can be solved by this method. The method can also solve other nonlinear partial differential equations.     

The relation among the hyperbolic-function-type exact solutions of nonlinear evolution equations

First, we investigate the solitary wave solutions of the Burgers equation and the KdV equation, which are obtained by using the hyperbolic function method. Then we present a theorem which will not only give us a clear relation among the hyperbolic-function-type exact solutions of nonlinear evolution equations, but also provide us an approach to construct new exact solutions in complex scalar field. Finally, we apply the theorem to the KdV–Burgers equation and obtain its new exact solutions.     

New explicit and exact solutions for a compound KdV-Burgers equation

In this paper, new explicit and exact solutions for a compound KdV-Burgers equation are obtained using the hyperbolic function method and the Wu elimination method, which include new solitary wave solutions and periodic solutions. Particularly important cases of the equation, such as the compound KdV, mKdV-Burgers and mKdV equations can be solved by this method. The method can also be applied to solve other nonlinear partial differential equation and equations.     

A new complex line soliton for the two-dimensional KdV–Burgers equation

Making use of a extended tanh method with symbolic computation, we find a new complex line soliton for the two-dimensional (2D) KdV–Burgers equation. Its real part is the sum of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D KdV (KP) equation, and its imaginary part is the product of the shock wave solution of a 2D Burgers equation and the solitary wave solution of a 2D MKdV (MKP) equation.     

KdV型方程孤波解与KdV-Burgers型方程行波解的稳定性

证明了KdV型议程的孤波解和KdV-Burgers型方程的行波解在李亚诺夫意义下是不是稳定的，从而修正了文献中的一些结论。     

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.     

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.     

Consistent Riccati expansion solvability,symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves

We study a forced variable-coefficient extended Korteweg-de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.     

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.