Study On Symmetries,Conservation Laws And Exact Solutions Of Some Mathematical Physical Equations

With the rapid development of science and technology,a large number of mathematical physical equations have appeared in the fields of physics,engineering,economy etc,and many natural phenomena,physical phenomena,mechanical problems can be described by these mathematical physical equations,which provides us with a reliable physical backgrounds and practical significance for the study of mathematical physical equations.Symmetries,conservation laws and solutions play an significant role in the study of mathematical physical equations.Symmetries reflect the laws of the structure of mathematical physical equations,conservation laws reflect the characteristics of motions and changes of mathematical physical equations,the solution reveals the change of physical properties of the equations.Over the past few decades,many mathematical physical enthusiasts actively devoted themselves to studying the related properties of mathematical physics equations,and they derived many effective methods to construct various symmetries,conservation laws and exact solutions,and obtain some strong and effective conclusions.However,there is still a lack of research on the relationship between the symmetry,conservation laws and solutions of mathematical physical equations.Based on the previous work,it is expected to find new algorithms and applications of symmetry,conservation laws and exact solutions of mathematical physical equations.The main contents of this paper are as follows:In Chapter 1: The research backgrounds and development of symmetries,conservation laws,exact solutions and mathematical mechanization of mathematical physical equations are briefly introduced.In Chapter 2: The basic idea of symmetry /adjoint symmetry pair method and Ibragimov's new conservation theorem are briefly introduced.Then,the conservation laws of two kinds of shallow water wave equations are successfully constructed with the help of symbolic computing techniques of Maple and Mathematica,and the above two methods are compared through obtained results.In Chapter 3: The basic idea of using power series polynomial form method to solve the symmetries of mathematical physical equations is introduced,and the power series form symmetries of one-dimensional Green-Naghdi water wave equations are successfully solved by using this method.Then,the sub-algebra of the system and its optimization system are calculated,and the similar solutions is obtained by similarity reductions.In Chapter 4: We introduce the basic idea of transforming a single-domain soliton equation into the n-coupled mathematical physical equations by thoughing a special transformation,and the conservation laws of two kinds of n-couple systems are derived by direct method.Finally,the soliton solutions of the 2-coupled Gardner equations are derived by Hirota bilinear method.In Chapter 5: Based on conservation laws and differential constraits conditions and Riemann invariant,some solutions and Riemann invariants of several equations are constructed.In Chapter 6: The main research works of this paper are concluded and the future development directions are prospected.