Research On The Bifurcation And Dynamical Behavior Of Traveling Wave Solutions For Non-linear Wave Equation

The Nonlinear wave equation and bifurcation theory of dynamical system arethe hotspot in the nonlinear science, more and more scientists throw theirself intothese areas. In the paper, from the viewpoint of dynamical system, we investigatedthe traveling wave solutions, the bifurcations of traveling wave solutions and theirdynamical behavior of the nonlinear wave equation. In the mean time, combining thesymbolic algebra of computer and analysis of phase portraits, we obtained thedifferent types of traveling wave solutions between different equations and analyzedthe reason why these complex traveling wave solutions arise. In the real model,traveling wave solutions with boundary have strongly practical value, but the methodwhich find the traveling wave solutions can not ensure whether the solutions isbounded. In this paper, according to the theory of dynamical system, we investigatethe explicit and exact traveling wave solutions of nonlinear wave equation by usingcharacter of the closed trajectory connecting equilibrium points and the relationbetween the obits and traveling wave.When the wave equation has dissipative term, in some case, the exact travelingwave solutions can not be obtained, so we began to study the approximate methodwhich find the solutions. In the chapter 3, by using the dynamical behavior of orbitsnear the equilibrium points and the method of symbolic algebra of computer,omitting some higher order term, we can obtain the explicit expression ofapproximate solutions, so these enrich the method which study the approximatesolutions.When we study the nonlinear wave equations, a lot of wave equation existnon-analytical (non-smooth) solutions. For these non-analytical traveling wavesolutions, for example, especially, Peakon and Compacton, we explain the reasonwhy these non-analytical traveling wave solutions arise, and prove that peakon isgeneralized solution in the sense of generalized differential and the compacton is weak solution. In the meantime, we open out the relation between the non-analyticalwave solutions and analytical wave solutions, and the condition which bifurcation ofdifferent traveling wave solution arise.Moreover, in this paper, we prove that the nonlinear wave equation can givebirth to compacton when the integral constant is not 0. Especially, when the firstintegral exists logarithm function, the solutions of wave equation are very complex,and the wave equation may produce non-numerable compacton which the energy ofthese compacton have a maximum.Finally, we investigate the traveling wave solutions, bifurcations of its andthe their dynamical behavior of the hydrogen-bounded system in the phase cylinder,show that there exists rotation periodic wave solutions and breaking wave anddisplay the bifurcation which these traveling wave arise. In the paper, we alsoexplain these waves arising.