Gui Mu

New Two-soliton and Periodic Solutions to KdV Equation

There has been an active and exciting search for explicit solutions of nonlinear evolution equations(NLEES) ever since the discovery of the soliton in 1834.Many method have been proposed over these years for finding these solutions [1-10] including a number of asymptotic methods that have been proposed recently by J.He [11,12] and the double-exp function method [13] for finding the exact multi-wave solutions and double-wave solutions of NLPDEs. In this work, we show that the most elementary ansatz of double Exp-function method can be produced by an extension of two-soliton ansatz in a fractional form. Applying this ansatz to KdV equation, new abundant two-solitary solutions and periodic solution are obtained. We consider KdV equation [1]:

Analytic multi-soliton solutions of the generalized Burgers equation

The most elementary ansatz of the double-Exp-function method for finding exact double-wave solutions can be produced by an extension of a two-soliton ansatz in a fractional form. The generalized Burgers equation is used as an example, and closed form analytic multi-soliton solutions are obtained for the first time.

Explicit doubly periodic soliton solutions for the (2+1)-dimensional Boussinesq equation

In this paper, (2+1)-dimensional Boussinesq equation is investigated. New explicit two soliton solution, periodic solitary wave solution and doubly periodic soliton solution are obtained by using special transformation of unknown function and three wave method with the aid of Maple.

Analysis of Stability of Traveling Wave for Kadomtsev-Petviashvili Equation

This paper presents the boundedness and uniform boundedness of traveling wave solutions for the Kadomtsev-Petviashvili (KP) equation. They are discussed by means of a traveling wave transformation and Lyapunov function.

Combined Exp-Function Ansatz Method and Applications

Our aim is to present a combined Exp-function ansatz method. This method replaces the traditional assumptions of multisolitons by a combination of the hyperbolic functions and triangle functions in Hirota bilinear forms of nonlinear evolution equation. Using this method, we can obtain many new type analytical solutions of various nonlinear evolution equations including multisoliton solutions as well as breath-like solitons solutions. These solutions will exhibit interesting dynamic diversity.

Spatio-Temporal Feature in Two-Wave and Multi-Wave Propagations

It is well known that the wave propagation depends mainly on its velocity and frequency in one direction for a single wave[1-3]. There are many literatures devoted researches of single wave propagation such as solitary wave, periodic wave, chirped wave, rational wave etc [4-6]. However, what can be happen when two and even more waves with different features propagate together along different directions? In the past decades, many methods have been proposed for seeking two waves and multi-wave solutions to nonlinear models in modern physics. Recently, some effective and straight methods have been proposed such as homoclinic test approach(HTA)[7-8], extended homoclinic test approach(EHTA)[9-10] and three wave method [11-12]. These methods were applied to many nonlinear models. Several exact waves with different properties have been found out, such as periodic solitary wave, breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave, kinky kinkwave, periodic kinkwave, two-solitarywave, doubly periodicwave, doubly breather solitary wave, and so on. Because of interaction between waves with different features in propagation process of two-wave or multi-wave, some new phenomena have been discovered and numerically simulated, for example, resonance and non resonance, fission and fusion, bifurcation and deflexion etc. Furthermore, similar to the bifurcation theory of differential dynamical system, constant equilibrium solution of nonlinear evolution equation and propagation velocity of a wave as parameters are introduced to original equation, and then by using the small perturbation of parameter at a special value, two-wave or multi-wave propagation occurs new spatiotemporal change such as bifurcation of breather multi-soliton, periodic bifurcation and soliton degeneracy and so on.

Notice of Retraction Analytic muti-solitary solution of some generalized nonlinear equation

The most elementary ansatz of the double Expfunction method for finding exact double-wave solutions can be produced by an extension of two-soliton ansatz in a fractional form. Applying this method to generalized Burgers-Fisher equation and generalized Burgers-Huxley equation, closed form analytic muti-solitary solutions are obtained for the first time.

Multi-soliton Solutions to Generalized B(n,n) Equtions

The double Exp-function method is applied to the generalized Burgers equations, and one soliton and multi-soliton solutions are obtained. Result shows that the double Exp-function method is effective and straight for finding exact double-wave solutions of nonlinear evolution equations.