Inverse Scattering Transform For The Non Isospectral Equations With Self-consistent Sources

In the paper, by the Inverse Scattering Transform(IST), solutions of some integrable equations with self-consistent sources are obtained. These equations are the AKNS hier-archy with self-consistent sources, the non-isospectral KdV hierarchy with self-consistent sources, the non-isospectral mKdV hierarchy with self-consistent sources and the non-isospectral nonlinear Schrodinger(NLS) hierarchy with self-consistent sources. Dynamics of some obtained solutions are investigated. Reductions of the AKNS hierarchies with self-consistent sources are considered from the uniqueness of Jost functions. The isospectral and non-isospectral AKNS hierarchies with self-consistent sources and their solutions are reduced to the mKdV hierarchies with self-consistent sources, the sine-Gordon(SG) hier-archies with self-consistent sources and the NLS hierarchies with self-consistent sources. The infinitely many conservation laws of some hierarchies with self-consistent sources are derived from Lax pairs.In Chapter 2, for completeness, we derive out isospectral and non-isospectral AKNS hierarchies with self-consistent sources, and repeat the IST procedure for isospectral case. In Chapter 3, we derive some classical non-isospectral hierarchies including the nonisospec-tral mKdV hierarchy with self-consistent sources and the nonisospectral NLS hierarchy with self-consistent sources. Then we derive their solutions through the IST and also investigate dynamics of some solutions. In Chapter 4, the reductions are considered by making use of the uniqueness of Jost functions. The AKNS hierarchies with self-consistent sources and their solutions are reduced to the mKdV hierarchy with self-consistent sources, the SG hierarchy with self-consistent sources and the NLS hierarchy with self-consistent sources and their solutions. Dynamics of some solutions are investigated. In Chapter 5, we derive out the nonisospectral KdV hierarchy with self-consistent sources. Solutions are obtained by the IST. Nonisospectral dynamics, including 1-soliton characteristics, two solitons scattering and special behaviors related to sources (for example, the "ghost" solitons in degenerate 2-soliton case), are investigated analytically. In Chapter 6, the infinitely many conservation laws of some hierarchies with self-consistent sources are de-rived from Lax pairs. The examples include isospectral AKNSSCS hierarchy, isospectral Kaup-Newell(KN) hierarchy with self-consistent sources, isospectral Ablowitz-Ladik(AL) hierarchy with self-consistent sources and isospectral KP hierarchy with self-consistent sources.