Research On Traveling Wave Solutions For Several Kinds Of Higher Order Nonlinear Wave Equations

In the paper, by using qualitative theory of diferential equation, the method ofbifurcations of dynamical systems, symbolic computation, numerical simulation and so on,we investigate the exact traveling wave solutions, the bifurcation phase portraits and therelations of the traveling wave solutions of higher order nonlinear wave equations. Initially,by using traveling wave transformation, the traveling wave equation can be turned into aplanar dynamical system. Subsequently, according to the theory of dynamical system, weinvestigate the explicit and exact traveling wave solutions of nonlinear wave equation byusing character of the closed trajectory connecting equilibrium points and there relationbetween the obits and traveling wave, and we get a variety of new exact solutions ofnonlinear wave equations.The major works of this dissertation mainly are as follows.In Chapter2, we use the bifurcation method of dynamical systems to study the pe-riodic wave solutions and their limits for the generalized KP-BBM equation. A numberof explicit periodic wave solutions are obtained. These solutions contain smooth peri-odic wave solutions and periodic blow-up solutions. Their limits contain periodic wavesolutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, andsolitary wave solutions. Compared to the previous research of the equation, most of thesolutions we obtain are new, which extends previous results in some extent.In Chapter3, by using the bifurcation method and qualitative theory of dynamicalsystems, we study the traveling wave solutions and their relations for quadratic nonlinearKlein-Gordon equation. Through some special phase orbits, we obtain many smoothperiodic wave solutions and periodic blow-up solutions. Their limits contain periodicwave solutions, blow-up wave solutions and solitary wave solutions.In Chapter4, we carry out the bifurcation analysis of the Klein-Gordon equationwith power law nonlinearity. Initially, the phase portraits will be obtained and the cor-responding qualitative analysis will be discussed. Several interesting properties of thesolution structure will be obtained based on the parameter.In Chapter5, we carry out the bifurcation analysis of the Klein-Gordon equationwith dual power law nonlinearity. Initially, the phase portraits will be obtained and thecorresponding qualitative analysis will be discussed. Subsequently, we present the rela-tions between the traveling wave solutions and the Hamiltonian h. Finally, we obtainedan implicit solution in terms of Gauss’ hypergeometric functions of the equation.In Chapter6, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey-Stewartson equation. A number of traveling wavesolutions are obtained. Those solutions contain explicit periodic wave solutions, periodicblow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions,and solitary wave solutions. Relations of the traveling wave solutions are given. Someprevious results are extended.