The Study On Conservation Laws And Exact Solutions Of Some Nonlinear Differential Equations

With the rapid development of science and technology,the application of nonlinear science in mathematical physics is becoming more and more popular.Therefore,it is necessary to study nonlinear differential equations.Conservation laws and exact solutions of nonlinear differential equations are a hot issue in the field of mathematical physics.It is of great significance to discuss the integrability and linearization of nonlinear differential equations.Based on the symmetry method,combining with the symbolic computation system MATHEMATICA and Wu's method,we analyze the classical Lie symmetry and ?-symmetry of several nonlinear differential equations and the conservation laws and exact solutions are constructed.The first chapter,we introduce the development background,research status,significance and related knowledge of Lie symmetries theory,?-symmetry,conservation laws and exact solution methods briefly.The second chapter,we study the nonlinear Jaulent-Miodek(JM)equations with the classical Lie symmetry method.The classical Lie symmetry and sub-algebraic onedimensional optimal system of equations are obtained.Based on the adjoint equation method proposed by Ibragimov,the conservation laws of the JM equations are given.Finally,the exact solutions of the JM equations are given by using the power series method and the conservation law respectively.The third chapter,we analyze the(3+1)-dimensional KP equation by Lie symmetry and its one-dimensional optimal system and partial conservation law are given.Then,the new exact solutions of the KP equation are given by the conservation law.The fourth chapter,with the help of symbolic computing system,the ?-symmetry of three nonlinear partial differential equations is studied.Then,we find new invariant solutions of the equation by using ?-symmetry.The fifth chapter,the contents of this thesis are summarized and the future research contents are prospected.