Research On The Bifurcations Of Traveling Wave Solutions For Several Classes Of Nonlinear Wave Equations

In this thesis, from the viewpoint of bifurcation theory of dynamical systems, weinvestigate the bifurcations of several classes of nonlinear wave equations. By makingfull use of the first integrals and phase portraits of integrable wave systems, we study theexplicit and exact traveling wave solutions of the nonlinear wave equations. Meanwhile,by using the qualitative theory of di?erential equations, we make an analysis of theexistence of smooth and non-smooth traveling wave solutions which are di?cult toobtain. This thesis consists of six chapters.In Chapter l, we summarize the historical background, research developments andsignificance of nonlinear wave equations.In Chapter 2, we mainly introduce the approach of dynamical systems proposed byProfessor Jibin Li for studying singular nonlinear wave equations. We call the methodas"three-step method".In Chapter 3, the bifurcations and dynamical behavior of the nonlinear dispersionDrinfel'd-Sokolov (D(m,n)) system is studied by using the three-step method. Aftermaking a transformation of time scale, the singular traveling wave system of D(m,n)system is reduced to a regular dynamical system. Under di?erent parametric condi-tions, various su?cient conditions to guarantee the existence of smooth and non-smoothtraveling wave solutions are obtained by using the relation between the singular systemand the regular system. And how smooth period traveling wave solutions lose theirsmoothness and become non-smooth period traveling wave solutions is explained.In Chapter 4, the dynamical behavior of the generalized Camassa-Holm-KP equa-tions is studied by the three-step method. The existence of solitary wave solutions,kink and anti-kink wave solutions, compacton solutions, infinitely many smooth andnon-smooth traveling wave solutions are proved. Under di?erent regions of parametricspaces, various su?cient conditions to guarantee the existence of above solutions aregiven. Some exact explicit parametric representations of the above waves are given.In Chapter 5, the theory of dynamical systems is used to investigate the exacttraveling wave solutions of the shallow long wave approximate equations. In di?erentregions of the parametric space, su?cient conditions to guarantee the existence ofsmooth solitary wave solutions, kink and anti-kink wave solutions and periodic wavesolutions are given. Some exact explicit expressions of the above waves are obtained.Finally, the summary of this thesis and the prospect of future study are given.