Exact Travelling Wave Solutions And Their Bifurcations For Some Nonlinear Wave Equations

In this dissertation,the approach of dynamical system is used to discuss the ex?act travelling wave solutions and their bifurcations for some nonlinear wave equations,which have important applications in mathematics and physics.These equations include Raman soliton equations,as well as certain coupled nonlinear equations,ion-acoustic wave models and higher-order nonlinear equations.The rich dynamical behaviors and the bifurcations with the parameters of the traveling wave systems corresponding to these nonlinear equations are analyzed in detail.The exact parametric representations of different traveling wave solutions are obtained by some complex calculation.Raman soliton model in nanoscale optical waveguides,with metamaterials,having Kerr-law nonlinearity and parabolic-law nonlinearity are investigated by the method of dynamical systems and bifurcations,respectively.The exact travelling wave solutions and their bifurcations for these equations are discussed.Because the functions ?(?)in the solutions q(x,t)=?(x-vt)exp(i(-kx+?t))satisfy a singular planar dynamical system having two singular straight lines.By using the bifurcation theory of dynami-cal systems to the equations of?(?)under 23 different parameter conditions,bifurca-tions of phase portraits and 92 different exact traveling wave solutions including solitary wave solutions,periodic wave solutions,kink and anti-kink solutions,periodic peakons,peakons as well as compactons for the system are given.In the case of parabolic-law nonlinearity,it is much more difficult to study the exact travelling wave solutions and their bifurcations for the equation including extra quartic nonlinear item.According to the bifurcation theory of dynamical systems,the system is investigated more carefully,the 28 representative phase portraits are drew,and 62 different traveling wave solutions of the system such as solitary wave solutions,periodic wave solutions,kink and anti-kink solutions,periodic peakons,peakons,pseudo peakons and compactons as well as their exact parameter expressions are obtained.In addition,some coupled nonlinear equations,ion-acoustic wave models and high-order nonlinear equations are studies respectively.For the coupled nonlinear equations system,its travelling wave system is the first class singular traveling wave system de-pending on 9 parameters.By using the bifurcation theory and the method of singular traveling wave systems,it is showed that there exist parameter groups such that this sin-gular system has kink and anti-kink wave solutions,periodic wave solutions,periodic peakons and compactons as well as different solitary wave solutions.For the three ion-acoustic wave models which are governed by three partial differential equation systems respectively,their travelling wave equations also are the first class singular traveling wave systems depending on different parameter groups.By studying the bifurcations of these dynamical systems,it is showed that there exist parameter groups such that these singular systems have solitary wave solutions,periodic wave solutions,pseudo peakons,periodic peakons,as well as different compactons,which complete the stud?ies of the three papers[1-3].Lastly,for the five high-order nonlinear equations,the exact traveling wave solutions are studied by using the theory of dynamical systems.Based on Cosgrove's work,infinitely many soliton solutions and quasi-periodic solu-tions are presented in an explicit form.The existence of uncountably infinite many double-humped solitary wave solutions is proved,and the parameters range as well as geometrical explanation of solitary wave solutions are discussed.