The Symmetry Reduction Of The Initial Value Problem And The Group Classification Of Nonlinear Evolution Equations

Nonlinear science is substantially studied and widely applied in natural sciences such as chemistry, mathematics, biology, physics, economy and so on. There are many nonlinear systems. In the research process, we encounter a wide variety of nonlinear partial differential equations (PDEs). The method to solve the equations and the properties of their solutions are important parts of nonlinear science. Up to now, many methods have been established and developed to solve the nonlinear PDEs. It is well known that symmetry group theory is one of the effective ways in studying exact solutions of nonlinear PDEs.As the study of the nonlinear systems developing, it is found that many PDEs in application depend on a small parameter, which are usually called perturbed (approximate) PDEs, we need to seek their approximate solutions. Based on the theories frame of the unperturbed equations, some related anal-ysis have been done on the perturbed equations and many efficient conclusions have been achieved. Some symmetry perturbation methods based on the Lie theory have been established.In the first chapter, we study the background, current situation and basic notation and theory of our research work, at the same time introduce the main work of this paper.In the second chapter, classification and symmetry reduction of initial value problems for third-order evolution equation ut= F(u, ux)uxxx+G(u, ux, uxx), F(u, ux)≠0 and fourth-order evolution equation ut= -F(u)uxxxx-G(u, ux)uxxx+H(u, ux, uxx), F(u)≠0 are discussed by generalized conditional symmetry method,and the solutions are obtained.In the third chapter,classification and symmetry reduction of initial value problem for nonlinear systems of PDEs ut=f(u,v)uxx+g(u,v),f(u,v)≠0 vt=p(u,v)vxx+q(u,v),p(u,v)≠0 is successfully presented by a developing generalized conditional symmetry method.In the fourth chapter,we use the approximate generalized conditional symmetry method to research the classification and symmetry reduction of initial value problem for perturbed KdV-Burger equation ut=guxxx+(?)G(u)uxx+F(u,ux),g≠0 and perturbed diffusion-reaction type equation ut=(A(u)ux)x+(?)B(u)ux+Q(u) and develop the method from unperturbed case to perturbed one.In the fifth chapter,we discuss the preliminary group classification for one wave equation utt=f(x,u,ux)uxx+g(x,u,ux).This dissertation end with summary and research prospects.The innovations and features of this dissertation are as follows:(1)In research objection,we discuss the nonlinear PDE or systems of non-linear PDE which have physical background,such as from one single equation to system of equation and from unperturbed equation to perturbed equation. (2)In the method, we successfully develop the method to initial value problem for system of equation and put forward the method to deal with the initial value problem for perturbed equation.(3)In content, we research the classification and symmetry reduction of initial value problem for higher-order evolution, systems of PDE and per-turbed equation, and obtain the analytical solutions of initial value problem which offer important information for further study these equations. At the same time, we study the preliminary group classification for one class of wave equation.