Symmetry Reductions And Conservation Laws Of Several Nonlinear Equations

With the booming development of the nonlinear science,a large amount nonlinear differential equations emerge in physics and natural sciences fields and so on.Solving differential equations for further studies of the phenomenon undering the equations,is one of the core issues studied by mathematicians and physicists.In this paper,based on the method of Lie symmetry and the computer system Maple,we reduced the dimension of several kinds of differential equations,and several types exact solutions are obtained.In this paper,we mainly study from the following several aspect:In chapter 1,we briefly introduce the research background and development of soliton theory,symmetry theory and symbolic computation,meanwhile,the contribution of Chinese scholars.In chapter 2,the basis content of the method Lie symmetry and conservation laws are introduced,which are useful for us to seek the exact solutions and conservation laws of the equations.In chapter 3,with the help of Maple,we studied two kinds of nonlinear partial differential equations by means of the method of symmetry,several reduced equations and exact solutions with arbitrary functions are obtained,and we displayed the figures of some solutions,which plays a important role for us to analysis the form and the movement trend of the solutions.In chapter 4,we obtain the reduced equations and exacted solutions of the generalized Boussinesq equation by means of the method of symmetry,several kinds of method to construct conservation laws are introduced,and some conservation laws of the generalized Boussinesq equation are obtained with the help of these method.In chapter 5,we summarized the work of this paper and the prospect for further work.