Constraints And Solutions For Soliton Equations

The dissertation mainly consists of the following:1. We review straight ways and limit procedures to get limit solutions via many solving methods including Inverse Scattering Transform, Darboux transformation, Hirota mathod and Wronskian technique. We also introduce limit symmetry consisting of wave functions, taking the KdV equation as an example.2. We study symmetry constraints related to squared eigenfunctions(or wave functions) for the KdV equation, the modify KdV equation and the sine-Gorden equation. Then we solve these symmetry constraints by means of hirota method and get N-soliton solutions. Then these solutions in Wronskian form are verified. We also study the symmetry constraint of KdV equation related to limit symmetry and get 1-soliton solution for the KdV equation.3. We give a equation with sources related to the above limit symmetries. This equation is Lax integrabe and its 1-soliton solution and 2-soliton solution are obtained by using Hirota method. These obtained solutions are shown to be double-pole solutions of the KdV equation with self-consistent sources (the sources will be changed to the limit one in the limit prosedure). We then derive N-soliton solution from the KdV equation with self-consistent source by limit procedure. Further the multi-pole solution of the KdV equation with self-consistent source is presented.4. The modified Kadomtsev-Petviashvili hierarchy with self-consistent sources is derived from a generalized dressing operator and the corresponding wave functions. This hierarchy includes two types of the modified KP equation with self-consistent sources and admits reductions to k-constrained the modified KP hierarchy and to modified Gelfand-Dickey hierarchy with sources.5. The nonisospectral Toda hierarchy is solved by inverse scattering transform.