Research On Application Of Lie Groups At Integrability Of Nonlinear Differential Equation

This article studied method to utilize invariant manifold and one-parameter Lie groups to constitute first-integral of autonomous system; studied adjoint Lie groups system of general three body and conserved quantity and increased Lie groups and conserved quantity against special case;studied method to utilize one-parameter Lie groups to acquire invariant solution of partial differential equa-tion.Main content and result of study includes following several aspects:1) It is a few methods which are used to constructed First Integral with Time Variable by means of invariant manifolds and their cluster that are discussed in the paper.2)This paper focuses on adjoin Lie group system and conservation variable of three-body on the general condition, at the same time, three-body expansion adjoin Lie group system was achieved by virtue of special power potential function. Furthermore, using the three-body expansion adjoin Lie group system, a lot of conservation variables were attained except the 10 classic conservation variables.3)In this paper,algebraic curve solution about Travelling Wave solution equation of Fisher equation is presented by the exactly divisible theory of binary polynomial on the complex field . Liouville Integrable Condition about the equation is obtained and proved.4)Adjoint One-parameter Lie Group for a Class of nonlinear wave equations are obtained,and are used to reguce some ordinory differential equations,Further,a new class of invariantly exact solutions including travelling wave solution are obtained in some Special parameters.