Studies On 1+1 Dimensional Soliton Equationswith Self-consistent Source

The soliton equations with self-consistent sources(SESCS) have wide applications in physics. In recent years, how to solve these equations and how to construct B¨acklund transformation between equations have become two important topics in soliton theory and integrable system. Our dissertation focuses on the above two questions.There are three parts in this dissertation. The speci?c results are given as follows:1. Blow-up solutions to the complex Korteweg-de Vires equation with self-consistent sources. Under the assumption of space variable being real and of the wave number being complex, Blow-up solutions to the complex Korteweg-de Vires equation with self-consistent sources(Kd VSCS) are considered. Blow-up solutions are derived from known 1-soliton solution and 1-negaton solution of the complex Kd VSCS, respectively. In addition, dynamics of the solutions are also illustrated. Compared with the blow-up solution of Kd V equation,the sources may change the orbit of blow-up points.2. The q-deformed Korteweg-de Vires hierarchy with self-consistent source(q-Kd VSCS),Darboux-B¨acklund transformation, soliton solution. Auto-Darboux-B¨acklund and non-autoDarboux-B¨acklund transformation are ?rstly constructed for q-Gelfand-Dickey hierarchy with self-consistent source. Secondly, through this non-auto-Darboux-B¨acklund transformation, soliton solutions for the ?rst type of q-Kd VSCS are obtained.3. B¨acklund transformation for 1+1 dimensional soliton equations with self-consistent sources. B¨acklund transformation between Kd VSCS and m Kd VSCS, m Kd VSCS and HDSCS are given, respectively.