Some Study On Nonlinear Partial Differential Equations And Integrable System

Nonlinear partial differential equations and integrable system are very important in the study of applied mathematics and mathematical physics, which application is very extensive, and become hot spot in interdisciplinary researches.This paper focuses on the generalized conditional symmetry of the diffusion equation with source term, and Painleve property and exact solutions of multi-component Sasa-Satsuma(MSS) equation. This diffusion equation has a wide range of applications in physics, diffusion process and engineering sciences and has been applied to describe several situations such as heat conduction by electrons in a plasma, heat conduction by radiation in a fully ionized gas and turbulent diffusion. MSS equation can be seen as the generation of normal Sasa-Satsuma equation, and the latter is usually used to describe many phenomena the propagation of an ultra-short pulse through optical fibres. Therefore it makes sense to study these equations. Generalized conditional symmetry approach plays an important role in seeking some exact solutions with special significance of nonlinear partial differential equation. And these exact solutions though this method can not generally be obtained by Lie point symmetry, so it is of great research value. Whether equations can pass Painleve test is an important condition of judging equations completely integrable. We can also obtain Backlund transformation, exact solutions, Darboux transformation, Lax pairs and so on by Painleve test, so it is meaningful to analyze Painleve property of equations.This paper first briefly introduces nonlinear partial differential equations, integrable system, and the background and research status of generalized conditional symmetry and Painleve analysis. Then we discuss the generalized conditional symmetry of the diffusion equation ut=e-qx(e-pxP(u)u")x+Q(x,u) with source term, and clarify the symmetries, and the equations we clarify can be reduced to dynamic systems or can be solved exactly. Ultimately we discuss the Painleve property of multi-component Sasa-Satsuma equation by means of WTC method, and then truncate the Laurent expansion, construct the corresponding exact solution and obtain new coherent structures.