Construction Of Lax Pairs And Darboux Transformations For Some Soliton Equations

Soliton theory is one of significant parts in applied mathematics and mathematical physics, and it has important applications in the non-linear physics, dynamics, superconductivity, quan-tum field theory, meteorology, communication, and so on. However, it is a difficult problem to seek exact solutions for the non-linear evolution equations all the time. The solving of non-linear equations can be converted to that of linear equations based on their Lax pairs. So far, prolongation structure theory is one of the successful methods for constructing Lax pairs of the non-linear evolution equations. The Darboux transformation based on the Lax pairs is one of efficient methods to derive the exact solutions of the non-linear evolution equations.This thesis is emphasized on non-linear evolution equations. Prolongation structure method to construct the Lax pairs of these equations and Darboux transformation method to solve these are discussed.The main works are as follows:Chapter1is concerned with the backgrounds, previous works and methods about the soliton theory, integrable systems, prolongation structure theory, Darboux transformation and Backlund transformation, respectively.In Chapter2, definitions of Lax pairs and Lax equations are introduced, then the Lax pairs of the Boussinesq equations and modified KdV equations are constructed based on the prolongation structure theory.In Chapter3, Darboux transformations of the classical Boussinesq-Burgers system, Boussi-nesq equations and complex KdV equation are constructed. The single soliton solution of the classical Boussinesq-Burgers system and exact solutions for the complex KdV equation based on the Darboux transformations are obtained.Finally, a brief summary of the thesis is given, and some future works are also pointed out.