Soliton-like Solutions For Nonisospectral Equation Hierarchies

In this paper, We obtain the solutions for nonisospectral evolution equation heirar-chies and r-sysmetry hierarchies by the inverse scattering transform(IST), Hirota method and Wronskian technique, we also investigate the properties of these solutions. In chapter 2, under the condition which spectral parameter k evolves according to time, we derive out equation hierarchies for KdV system, mKdV system, sine-Gordon system and AKNS system. Equation hierarchies for KdV system, mKdV system, sine-Gordon system and nonlinear Schrodinger system are reduced from equation hierharchy for AKNS system. Specially, isospectral hierarchies, nonisospectral hierarchies and related Ï„-sysmetry hierarchies are special cases of them. In chapter 3, by IST, we obtain N-soliton solution to equation hierarchy for KdV system, and reduce to N-soliton solution to isospectral KdV hierarchy, nonisospectral KdV hierarchy and Ï„-symetry hierarchy . In chapter 4, by IST, we obtain N-soliton solution to equation hierarchy for AKNS system, and reduce to N-soliton solution to equation hierarchies for KdV system, mKdV system, sine-Gordon system and nonlinear Schrodinger system. In chapter 5, we investigate the nonisospectral sine-Godorn equation, nonisospectral nonlinear Schrodinger equation by Hirota method and Wroskian technique, meantime we also consider Ï„-symetry equation for KdV, mKdV, sine-Gordon, nonlinear Schrodinger systems by Hirota method. The properties of these solutions are investigated. In chapter 6, we give gauge transform between nonisospectral AKNS hierarchy and isospectral AKNS hierarchy.