Integrability And Solution Of Schrodinger And Boussinesq-Burgers Equations

Nonlinear evolution equation is an important mathematical model for describing physical phenomenon and an important field in the contempary study of nonlinear physics, especially in the study of soliton theory. The research on the explicit solution and integrability are helpful in clarifying the movement of matter under the nonlinear interactivities and plays an important role in scientifically explaining of the corresponding physical phenomenon and engineering ap-plication. Many research topics, such as searching for exact explicit solutions, multi-soliton solution, the Painleve test etc., often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice, In recent years, the development of symbolic computation accelerates the research of nonlinear evolution equation greatly. Many new methods for constructing exact solutions of nonlinear evolution equations are proposed.This dissertation equations with the aid of symbolic computation, mainly studies Lax pairs and Painleve test of a kind of Schrodinger equation. And as for Boussinesq-Burgers equation, two types of N-fold Darboux transformations and some solutions are derived. The article consists of the following parts:Chapter 1 introduces the backgound, development and research method of soliton theory and gives Schrodinger equations a summery preparing for Chapter 4.Chapter 2 states the theory of "AC=BD" including babsic thought and its application describing other common research method.In Chapter 3, two kinds of integrability are introduced. One is Lax integrability including prolongation structure method for deriving Lax pair of equations. The other is Painleve inte-grability which contains Painleve test method, Backlund transformation and Darboux transfor-mation.Chapter 4 studies a kind of Schrodinger equation. Making use of symbolic computation we study Painleve integrability first, and then Lax integrability is discussed.Chapter 5 constructs the Darboux transformation of Boussinesq-Burgers equation. And on that base some exact solutions are presented.At last a short summery of the dissertation is given.