Studies On Exact Solutions Of Partial Differential Equations Based On Lie Symmetry Analysis

Partial differential equations (PDEs), also called mathematical physics equations, are derived from the models of Physics, Mechanics and Engineering, etc. The earlier PDEs, such as Laplace equation and Poisson equation are based on the Newton's theory, wave equation, heat equation, and so on. And these equations are all classical PDEs. From the 19th century, a lot of new PDEs are derived, including the famous Maxwell system, Schrodinger equation, Einstein equation, Yang-Mills equation, and reaction-diffusion equation, etc. More and more PDEs, especially the nonlinear equations and systems will be derived owing to the progress of the modern science and technology. In addition, the nonlinear wave equations are important mathematical models for describing natural phe-nomena and one of the forefront topics in the studies of nonlinear mathematical physics, especially in the studies of soliton theory. The research on finding explicit and exact solu-tions of nonlinear wave 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 corresponding 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 paper, the exact and explicit solutions, the dynamical behavior and the inte-grable properties for PDEs are investigated from the viewpoint of bifurcation theory of dynamical systems, Painleve analysis, (generalized) power series method, method of un-determined coefficient and a lot of special techniques, based on the Lie symmetry analysis. In detail, firstly, we solve the vector field or symmetries for a PDE with the Lie symmetry analysis method. Then, we reduce the PDE to the ordinary differential equations (ODEs) by the similarity transformations, it is a significant progress for dealing with the exact solutions of a PDE. That is, we transform a complicated PDE, including the nonlinear equation, variable coefficient equation, etc., to a ODE. Thirdly, we deal with the ODEs with the various method, such as the bifurcation theory of dynamical systems, Painleve analysis, (generalized) power series method, method of undetermined coefficient and a lot of special techniques. This is the outline for studying the exact solutions for PDEs. As we all know, by employing the Lie symmetry analysis, we can also consider the the integrable property for PDEs. But this is not the theme for this thesis, so we will not investigate it in detail.The major works of this thesis are as follows.Chapter 1 is the introduction. In this chapter, the historical background, research developments, main methods and achievements of partial differential equations especially the nonlinear wave equations are summarized. The discovery, corresponding research approaches and recent advance of Lie symmetry analysis and Painleve analysis methods are introduced. The relationship between partial differential equations and dynamical behavior, Painleve properties along with the study on exact solutions of partial differential equations by using the Lie symmetry analysis method are presented.Chapter 2 is the preparation of theory. In this chapter, some preliminary knowledge of dynamical systems, theory of Lie groups and Lie algebras, prolongation of vector fields and the other basic mathematical theory and main results are introduced. This chapter is the preliminary knowledge of the thesis. For the sake of succinctness, we outline the summaries only. If it is necessary to learn some contents in detail, please consult the references.In Chapter 3, the Lie symmetry analysis method is performed for the general Burg-ers'equation. This equation is a nonlinear wave equation, and it is of great importance in both theory and applications. The equation has various solitary wave solutions, such as the shock wave solutions, sparse wave solutions, etc., so it is important to study the wave problems in fluid dynamics and aerodynamics. For example, in the model equations of fluid dynamics, we have the linear Burgers' equation ut+aux=?uxx and the nonlinear Burgers'equation ut+[f(u)]x=?uxx, respectively. In particular, if f(u)=1/2u2,then the latter is ut+uux=?uxx.Under the certain conditions, we can get the exact solutions for the two equations. Thus, the fluid dynamical property can be comprehended. More-over, the Burgers'equation can be transformed into other important mathematical physics equations such as the heat equation, so it plays a significant role in the nonlinear science, fluid dynamics and engineering.Based on the Lie symmetry analysis, we get the group-invariant solutions, iterated solutions. Then, by using the similarity reductions, we transform the general Burgers' equation to ordinary differential equations (ODEs), some of them are nonlinear and non- autonomous quations. Next, we deal with these ODEs by using the power series method and some special techniques, the exact solutions for the ODEs are obtained, so the solu-tions of the general Burgers'equation are presented simultaneously. Some new solutions are obtained for the first time in this chapter.In Chapter 4, we investigate the extended mKdV equation. As is well known, the KdV equation is a famous shallow water wave equation, and it comes of the studies on the water wave problems. The KdV type equations can depict the various shallow water wave moment, and they play an important role in fluid dynamics. The modified KdV (mKdV) equation, on the other hand, has recently been discovered, e.g., to model the dust-ion-acoustic waves in such cosmic environments as those in the supernova shells and Saturn's F-ring, etc.By using Lie symmetry analysis method, the vector field for the extended mKdV equation is obtained, and the vector fields of the several special classical KdV-type equa-tions are presented simultaneously. Then, by using the method of dynamical systems for the extended mKdV equation, all the exact solutions based on the Lie group method will be given. Especially, the bifurcations and traveling wave solutions are obtained. To guar-antee the existence of the above solutions, all parameter conditions are determined. Fur-thermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlin-ear science. Note that the vector fields of the several special classical KdV-type equations obtained in this way are a part of the vector fields rather than its complete vector fields, respectively. This reflect the intricacy of the Lie symmetry analysis. One of the highlight of the chapter is the method of dynamical systems. By employing this method, we discuss the dynamical behavior and get the all traveling wave solutions for the equation.In Chapter 5, the Lie symmetry analysis and the generalized symmetry method are performed for a short pulse equation (SPE). The SPEs are nonlinear wave equations also, it can depict some peculiar waves. It is of great importance in comprehending some special wave problems that the SPEs and its solitary wave solutions are investigated in detail. At the same time, the SPEs are important mathematical physics equations, it play a significant role in engineering and mechanics.Note that this equation is not a common evolution equation, but a mixed one. This brings some difficulty for the Lie symmetry analysis method. The symmetries for this equation are given. For the traveling wave solutions, the exact parametric representations are investigated. To guarantee the existence of the above solutions, all parameter condi-tions are determined. Furthermore, the exact analytic solutions are obtained by using the power series method. One of the highlight of the chapter is that we use the method of undetermined coefficient to solve the symmetries for the equation. Then, for getting the exact explicit solutions, the method of parameters are employed. It is not obtained by the general method for these solutions obtained in this chapter.In Chapter 6, we use Lie symmetry group methods to study a pair of bond pricing equations. The variable-coefficient PDEs are derived from the mathematical physics and engineering problems at first. Now, many PDEs are derived from society, biochemistry, environmental protection, information science, communication, finance and securities, etc., along with the progress of society and the development of science and technology. Moreover, these PDEs are complicated and variable-coefficient usually. The variable-coefficient PDEs in this chapter are of importance in financial mathematics and financial engineering, and it play an important role in bond price as well.The symmetries and similarity transformations for the two equations are provided, and all exact solutions are obtained. The general case??2 are considered simultane-ously. We show that when the reduced equations are obtained, the generalized power se-ries method and special techniques can be used to find the exact solutions.When the inho-mogeneities of media and non-uniformity of boundaries are taken into account in various real physical situations, the variable-coefficient nonlinear evolution equations (NLEEs) can often provide more powerful and realistic models than their constant-coefficient coun-terparts in describing a large variety of real phenomena. On the other hand, the Lie sym-metry analysis are more complicated than the constant-coefficient equations. The other highlight of the chapter is the generalized power series method and the method of unde-termined coefficient are utilized for tackling the exact explicit solutions.In Chapter 7, the Painleve analysis and Lie symmetry methods are performed for the generalized KPP equation, Newell-Whitehead equation and its special case. These PDEs play an important role in nonlinear science and engineering, and they are mathematical models of many wave problems and mechanics. It is of importance also in biomathcmat-ics, etc.Firstly, the Painleve property and the symmetries are obtained. Meanwhile, the exact solutions generated from the symmetry transformations are considered. Especially, the exact analytic solutions are investigated by the power series method. In this chapter, we considered three nonlinear evolution equations, the Painleve analysis is the highlight for this chapter. By the Painleve analysis method, the Backlund transformations and truncated expansion are obtained. Moreover, by the truncated expansion, the exact solutions are investigated for the three equations.In Chapter 8, we summarize the main contents for the thesis firstly. Then, the points of innovation of this thesis are presented. At last, the prospect of further study for this direction are given.