Exact Solutions Of Nonlinear Evolution Equations And Their Dynamic Behavior Studies

With the development of science and technology, there are many nonlinear problems in natural and social areas, which arouses much concern. These problems are usually characterized by nonlinear evolution partial differential equations. The research on find-ing explicit and exact solutions of nonlinear evolution equations and on analyzing the qualitative behavior of solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions, explain the corre-sponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply, and promote greatly the development of engineering technology and related subjects such as physics, mechanics and applied mathematics.In this dissertation, the exact travelling wave solutions, the bifurcations and dynam-ical behavior of travelling wave solutions of the nonlinear evolution equations are inves-tigated from the viewpoint of bifurcation theory of dynamical systems. The major works of this dissertation mainly are as follows.In Chapter 1, the historical background, research developments, main methods and achievements of nonlinear evolution equations are summarized. The discovery, corre-sponding research approaches and recent advance of non-smooth waves are introduced. The relation between nonlinear evolution equations and dynamical systems along with the study on nonlinear evolution 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 Chapter 2, the exact travelling wave solutions of the n+1 dimensional double sine-Gordon(DSG) equation and double sinh-Gordon(DSHG) equation are studied by using the dynamical system approach. First, we study the dynamical behavior of DSG equation and DSHG equation on phase cylinder and phase plane respectively and obtain all pos-sible explicit exact travelling wave solutions of the two equations in different parametric regions. Second, we investigate the two equations by using three different transforma-tions. Under some transformations, the resulting travelling wave systems are singular. After making a transformation of time scale, the singular systems are reduced to the regu-lar system. And the qualitative behavior of orbits of the regular system can be obtained by using the classical bifurcation theory of dynamical systems. Hence the dynamical behav-ior of the solutions of the singular systems are achieved from singular perturbation theory and the relation between the singular system and the regular system. An important fact is obtained:under these transformations the travelling wave solutions of DSG and DSHG equations are smooth. By the analysis of phase plane we give all possible explicit exact travelling wave solutions of the DSG and DSHG equations under these transformations. Finally, after verifying one by one, we illustrate that the solutions of DSG and DSHG equations which we obtained by making transformations are included in those obtained by considering the original equation directly. In other words, the solutions obtained by using transformations are just changed in forms and the transformations do not change the dynamical behavior of the equations essentially. We can conclude that the dynamical system approach is an effective method to solve nonlinear evolution equations. The trav-elling wave solutions obtained by using dynamical system approach are comprehensive and the approach can not be displaced by other methods.In Chapter 3, the exact travelling wave solutions and dynamical behavior of the gen-eralized Calogero-Degasperis-Fokas (gCDF) equation are studied. After making a trans-formation of time scale, the singular travelling wave system of gCDF equation is reduced to a regular dynamical system. The singular travelling wave system and the regular trav-elling wave system have distinct time scales which cause their some corresponding orbits to have distinct dynamical properties. And the relationship of orbits between the regular system and the singular system is analyzed by using singular perturbation theory. And the fact is proved that the singular homoclinic orbit of the regular system is corresponding to smooth periodic orbit or homoclinic orbit of the singular system; the heteroclinic orbit of the regular system is corresponding to smooth homoclinic orbit or heteroclicic orbit of the singular system. It shows that singular line does not always result in non-smooth so-lutions. Furthermore, the reason of the occurrence of breaking wave is explained. Finally, we give all possible explicit exact travelling wave solutions of gCDF equation, which in-clude not only smooth solitary waves, kink waves, anti-kink waves and periodic waves but also breaking waves with peak type or valley type (breaking in two sides), breaking kink waves and breaking anti-kink waves(breaking in one side). The existence of singu-lar line makes the dynamical behavior of travelling wave solutions of nonlinear evolution equations be more complex and the dynamical system approach is the effective tool to study these complex and interesting problems.In Chapter 4, the exact travelling wave solutions of the generalized nonlinear deriva-tive Schrodinger equation and the high order dispersive nonlinear Schrodinger equation are studied. According to the physical background and by proper travelling wave trans-formations, the research on the two equations is reduced to the research on the same Hamiltonian system. Through complete and delicate discussion of the dynamical behav-ior of the Hamiltonian system, we obtain all possible envelope solitary wave, envelope kink (anti-kink) wave and periodic wave solutions of the two equations, which are more complete than those in other papers.In Chapter 5, the exact solutions of the nonlinearly dispersive Schrodinger equation, i.e. NLS(m,n) equation, are studied. The research on complex nonlinear evolution equa-tion is transformed to the research on a planar integrable system by proper transformation. The dynamical behavior of solutions of NLS(m,n) equation is studied systematically by using the classical theory and methods of planar dynamical systems. The reasons for ap-pearance of non-smooth periodic cusp patterns and breaking patterns are explained and the sufficient conditions to guarantee the existence of smooth and non-smooth solutions are obtained. The explicit and implicit parametric representation of some smooth enve-lope solitary patterns, envelope kink patterns, periodic patterns and non-smooth periodic cusp patterns are given.In Chapter 6, the summary of this dissertation and the prospect of future study are given.