Qualitative Analysis Of Some Nonlinear Wave Equations

Wave equation is a kind of important differential equation,which is used to describe various wave phenomena in nature,such as sound wave,light wave,electromagnetic wave and water wave,etc.In this paper,several types of nonlinear wave equations,including Camass-Holm equation,Schr(?)dinger equation and related equations,are qualitatively analyzed,and the existence of traveling wave solution,well-posedness and wave breaking phenomena are studied.Firstly,the solitary wave solutions of the Camassa-Holm-Kadomtsev-Petviashvili(Camassa-Holm-KP)equation are considered.By using the analysis of the phase space,the basic properties of the equilibrium point of Camassa-Holm-KP equation without time delay are obtained,and the existence of solitary wave solutions is obtained.Furthermore,by developing the geometric singular perturbation theory,the existence of solitary wave solutions for the Camassa-Holm-KP equation with time delay is proved.At the same time,by analyzing the ratio of Abel integral,the monotonicity result of the wave speed of the delayed Camassa-Holm-KP equation with nonlinear strength of 1 is obtained.Then,the solitary wave solutions of the coupled perturbed Schr(?)dinger equations are discussed.For the case without time delay,three special solitary wave solutions are obtained by using the method of ordinary differential equations.On this basis,considering the corresponding system with delay.Combining the invariant manifold theory and Fredholm theory,the invariant manifold of the delayed system is constructed,and the corresponding homoclinic orbits are obtained.Then,the existence results of the solitary wave solutions of the delayed coupled Schr(?)dinger equations are obtained.Finally,the local well-posedness and the wave breaking phenomena of the twocomponent Camassa-Holm system and the corresponding modified system are considered.By using the Kato’s theory,the local well-posedness of the solutions of two kinds of systems is established respectively.Moreover,the conditions for the wave breaking are given.