Research On Exact Solutions Of Several Types Of Partial Differential Equations

This thesis mainly combines Lie symmetry analysis method and generalized symmetry method,based on the principle of homogeneous balance,constructing the B(?)cklund transformation of equations flexibly,using power series method,hyperbolic tangent function expansion method,Painlevé test,generalized CK method,using auxiliary calculation software Maple to obtain the exact solutions of several types of nonlinear partial differential equations.The idea of this thesis is as follows: first,the theory of two symmetric analysis methods is used to obtain the vector field of these equations.Based on the adjoint representation of the vector field,the optimal system of the equation is obtained;according to the similar reduction,the more complex nonlinear partial differential equations are reduced to the relatively simple ordinary differential equations;Next,observe the characteristics of the particular ordinary differential equations,and choose appropriate methods to obtain the exact solutions of the ordinary differential equations,so that the exact solutions of the corresponding partial differential equations can be obtained based on the exact solutions of the ordinary differential equations;Finally,the infinite dimensional conservation law of the partial equation is given according to the conclusions proposed by Ibragimov.The first chapter mainly introduces the research results obtained by scholars at home and abroad in the field of exact solutions of partial differential equations.Some practical methods for solving partial differential equations are summarized,and the related theories for solving exact solutions of partial differential equations are briefly introduced.In Chapter 2,a type of fifth order dispersion partial differential equation is considered,based on the principle of homogeneous balance,B(?)cklund transformation of the equation is constructed directly,and the solution of the hyperbolic tangent function of the equation is obtained.A generalized symmetry method is used to analyze the equation,and the corresponding vector field is obtained,meanwhile the optimal system of the equation is constructed.Dealing with the symmetry reductions on the basis of the optimal system,using the power series expansion method and the hyperbolic tangent function expansion method,the corresponding forms of exact solutions are obtained respectively.Finally the conservation laws of infinite dimension are given according to the symmetry and the adjoint equation.In Chapter 3,we consider a generalized mKdV equations with variable coefficients.Based on balance homogenous principle,the exact solutions of the equation are given through B(?)cklund transformation.Then the Painlevé test is performed to prove it's integrability.The power series solution of the equation is obtained by using the generalized CK method and the power series method.In Chapter 4,the generalized Sawada-Kotera equation is studied based on classical Lie symmetry analysis.The vector field is extended appropriately to obtain the infinitesimal generator,the generator obtained is analyzed,and the optimal system of the equation is constructed.On the basis of the symmetry reduction,the power series expansion method is used to obtain several power series solutions of the equation.The last chapter summarizes the conclusions and points out the unsolved issues in the thesis.