Exact Solutions And Differential Invariants Of Some Nonlinear Partial Differential Equations

Nonlinear partial differential equation is an important branch of modern mathematics,and it's a very important concept in both theory and practical applications.This subject mainly studies the three main aspects of nonlinear partial differential equations:firstly,the exact solution is used.The Backlund transformation and nonlinear superposition principle of Riccati equation yields an equation's infinite sequence exact solutions;Secondly,it describe the conservation laws,by proving the equation is nonlinearly self-adjoint to construct a general formula of conservation laws.Finally,using the equivalent infinitesimal and equivalent moving frame method to study the differential invariants.This article mainly includes the following five contents:The first chapter firstly discusses the development of nonlinear partial differential equa-tions and several methods for solving exact solutions.Secondly,the development of Lie group is discussed.Then the development of differential invariants and its application in other fields are discussed.Finally,the main work of this artical is briefly explained.The second chapter mainly uses the variable transformation method and the first integral method to reduce a class of nonlinear partial differential equations into Riccati equations,and uses the Backlund transformation and nonlinear superposition principle of Riccati equation to obtain it's infinite sequence solitary waves and periodic solutionsThe third chapter takes the generalized variable coefficient Hirota-Sastsuma equations as an example.The symmetry of the equation is obtained by the classical Lie symmetry analysis method,and the equations are proved to be nonlinear self-adjoint,thus constructing the general formula of conservation laws.The fourth chapter mainly introduces two methods for solving the differential invariants of nonlinear partial differential equations.One is to use the equivalent infinitesimal method,the differential invariants and the corresponding invariant equations can be obtained by the equivalent algebras of the equations;The second is to use the equivalent moving frame method proposed by Olver to construct a suitable moving frame to obtain differential invariants and corresponding algebras.The fifth chapter mainly summarizes the full text and looks forward to the future research directions of the exact solutions,conservation laws and differential invariants of nonlinear partial differential equations.