The Bifurcations Of Nonlinear Waves For Kundu Equation

In this paper,typical nonlinear wave equation,i.e.the Kundu equation is studied by the qualitative theory of differential equations and the bifurcation method of dynamical systems.We obtain some new exact solutions and reveal some interesting kink wave bifurcation phenomena.At the same time,the concrete variation of the kink waves for those bifurcations phenomena are shown in the figures.The concrete research work of this dissertation as follows:In Chapter 1,firstly,we give a simple summary of the historical background and the status quo of solving methods of the nonlinear wave equations.Secondly,we introduce some of the need to use bifurcation method of dynamics systems and planar system of qualitative theoretical knowledge.Finally,we briefly present the major works of this dissertation.In Chapter 2,we consider the Kundu equation ut=iuxx?i(c3|u|2?c5|u|4)u??|u|2ux??u2ux via the dynamical systems theory.Under transformations u(x,t)=e?i?te?(x?ct)?(x?ct)=e?i?tei?(?)?(?)with?=x?ct?and ?'(?)=c/2????/4?2(?).Firstly,we show the bifurcation phases portraits of the corresponding planar system.Using some special phase orbits,we obtain ?(?).Under different parameter changes,?(?)shows different traveling waves.By the relationship of ?(?)and ?(?),we get ?(?),finally,we obtain the exact solutions u(?)represented by ?(?)and ?(?).On the basis of this,we study the characteristics of these solutions with different parameters.In the end,the example confirmed by software Mathematica is given.In Chapter 3,we study the nonlinear wave described by ?(?)in chapter 2 for further analysis.We discuss the bifurcation of kink wave expressed by ?(?)under the parameter changes,that is,the low-kink waves can be bifurcated from four types of nonlinear waves,tall-kink waves,symmetric solitary waves,blow-up waves and anti-symmetric solitary waves;the tall-kink waves can be bifurcated from two kinds of periodic waves expressed by trigonometric function and elliptic function respectively,and the concrete variation of those traveling waves are shown in the figures.