Bifurcation Solutions For Two Classes Of Nonlinear Wave Equations

Nonlinear wave phenomena are widely existed in nature.The models of these natural phenomena are nonlinear wave equations,whose exact solutions can well reflect the characteristics and evolution of these nonlinear wave phenomena.This thesis studies two classes of nonlinear wave equations: the generalized Gilson-Pickering equation and the generalized(2+1)-dimensional comformable fractional Schr (?)dinger system.Some new exact solutions of the above two classes of nonlinear wave equations are obtained by using the dynamical system method.This thesis is divided into four chapters.In the first chapter,the research methods,contents and main results of this thesis are introduced.In the second chapter,first,we transform the planar dynamical system of the generalized Gilson-Pickering equation into the corresponding regular system by using the traveling wave transformation and the theory of singular traveling wave equations.Then,the theory of planar dynamic systems is used to analyze the types of equilibrium points of the corresponding regular system,and the bifurcations of phase diagram under different parameter conditions are drawn.Finally,under different parameter conditions,the traveling wave solutions of the generalized Gilson-Pickering equation are obtained,including solitary wave solutions,uncountablely infinite many compacton solutions,smooth periodic wave solutions,non-smooth periodic cuspon solutions,peakon solutions,kink wave solutions and anti-kink wave solutions.In the third chapter,first,the generalized(2+1)-dimensional comformable fractional Schr(?)dinger system is transformed into the corresponding Hamilton system and its first integral by using the fractional order transformation.Then,the theory of planar dynamic systems is applied to analyze the types of equilibrium points of the corresponding Hamilton system,and the bifurcations of phase diagram under different parameter conditions are drawn.Finally,the expressions of traveling wave solutions of the system are obtained by integrating along different orbits,including solitary wave solutions,periodic wave solutions,singular periodic wave solutions,singular wave solutions,kink wave solutions and anti-kink wave solutions.In the fourth chapter,the whole text is summarized and further questions are proposed for further investigation.