Qualitative Research, Based On The Theory Of Dynamical Systems Of Nonlinear Waves

Nonlinear wave equations are important mathematical models for describing natural phenomena and are one of the forefront topics in the studies of nonlinear mathemati-cal physics, especially in the studies of soliton theory. The research on finding explicit and exact solutions of nonlinear wave equations and on analyzing the qualitative behav-ior of solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions, explain the corresponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply, and promote greatly the development of engineering technology and related sub-jects such as physics, mechanics and applied mathematics.In this dissertation, the qualitative behavior of travelling wave solutions of the non-linear wave equations are investigated from the viewpoint of dynamical systems theory. Firstly, the dynamical systems theory are used to finding exact solutions of nonlinear par-tial differential equations. On the other hand, by using dynamical systems theory, the qualitative behavior of travelling wave solutions to the several families of nonlinear wave equations from physical background are studied, and the rich dynamical properties of these nonlinear models are shown. Moreover, the orbital stability of periodic wave so-lutions for the Schrodinger equation and uniformly continuity of a Korteweg-de Vries equation with nonlinear dispersion is successfully analyzed.The major works of this dissertation mainly are as follows.In Chapter1, the historical background, research developments, main methods and achievements of nonlinear wave equations are summarized. The discovery, correspond-ing research approaches and recent advance of non-smooth waves are introduced. The relation between nonlinear wave equations and dynamical systems along with the study on nonlinear wave equations by using the theory of dynamical systems are presented. In the end of the chapter, some preliminary knowledge of dynamical systems and the basic mathematic theory and main results of the three-step method are introduced.In Chapter2, the qualitative theory of differential equations is applied to the K(2,2) equation with osmosis dispersion. Smooth, peaked and cusped solitary wave solutions of the osmosis K(2,2) equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these solitary wave solutions of the osmo-sis K(2,2) equation. Moreover, we study the single peak solitary wave solutions of the Fornberg-Whitham equation under the inhomogeneous boundary condition. The condi-tions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. We obtain all smooth, peaked and cusped solitary wave solutions of the Fornberg-Whitham equation and analyze their analytic and dynamical behavior.In Chapter3, we study the relationship between period and energy of periodic trav-elling wave solutions for the φ6field model. The various topological phase portraits with periodic annulus are given by using standard phase portrait analytical technique. Some analytic behaviors(convexity, monotonicity and number of critical periods) of the period functions associated to periodic waves are investigated. We prove that the period function has exactly one critical period under certain conditions. Moreover, the numerical simu-lation is made. The results show that our theoretical analysis agrees with the numerical simulatioa.In Chapter4, we study the existence and orbital stability of periodic wave solutions for the Schrodinger equation. The existence of periodic wave solution is obtained by us-ing the phase portrait analytical technique. The stability approach is based on the theory developed by Angulo for periodic eigenvalue problems. A crucial condition of orbital stability of periodic wave solutions is proved by using qualitative theory of ordinal dif-ferential equations. The results presented in this paper improve the previous approach, because the proving approach does not dependent on complete elliptic integral of first kind and second kind.In Chapter5, The qualitative theory of differential equations is applied to a Korteweg-de Vries equation with nonlinear dispersion. The parametric conditions of ex-istence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek.In Chapter6, the summary and the prospect of future study are given.