On The Ingegrable Coupled Generalization Of Dispersionless 2+1 Dimensional Soliton Hierarchies

In recent years,dispersionless systems and coupling systems have been the hot topics in the soliton and integrable systems.Our dissertation focuses on the coupled generalization of the 2+1 dimensional dispersionless integrable systems and their related solution.There are three parts in this dissertation.The specific results are given as follows:1.This part focus on the integrable coupled generalization of dispersionless B type KP(BKP)hierarchy.The symmetry constraint for dispersionless BKP hierarchy is firstly derived by taking dispersionless limit of that for BKP hierarchy.Then,based on the symmetry constraint for dispersionless BKP hierarchy,a new extended dispersionless BKP hierarchy is constructed.In addition,the integrability of this new extended dispersionless BKP hierarchy is proved by presenting its zero-curvature equation and the related conservation equation.From its zero-curvature equation,two types of dispersionless BKP equations with self-consistent sources together with their associated conservation equations are obtained.Hodograph solutions for the first type of dispersionless BKP equations with self-consistent sources are finally obtained.2.This part mainly aims at the integrable coupled generalization of 2+1 dimensional dispersionless Harry Dym hierarchy.The symmetry constraint for 2+1 dimensional dispersionless Harry Dym hierarchy is firstly derived by taking dispersionless limit of that for 2+1dimensional Harry Dym hierarchy.Then the 2+1 dimensional dispersionless Harry Dym hierarchy is generalized.Moreover,the hodograph solutions to the first type of dispersionless Harry Dym equations with self-consistent sources are given.Finally,B¨acklund transformation between the extended dispersionless mKP hierarchy and extended 2+1 dimensional dispersionless Harry Dym hierarchy are also constructed.3.This part focus on the integrable coupled generalization of 2+1 dimensional dispersionless Toda lattice hierarchy.The symmetry constraint for 2+1 dimensional dispersionless Toda lattice hierarchy is firstly derived by taking dispersionless limit of that for 2+1 dimensional Toda lattice hierarchy.Then the 2+1 dimensional dispersionless Toda lattice hierarchy is extended and the hodograph solutions for the dispersionless Toda lattice equations with self-consistent sources are finally obtained.