The Exact Solutions To The Hierarchy Of Nonlinear Evolution Equation

This dissertation mainly researches the exact solutions of three hierarchy of nonlinear evolution equation:the first one is a hierarchy of inverse KdV equation. The general-ized Wronskian solutions are obtained and then the soliton solutions and rational solutions are presented. The second one is the Ragnisco-Tu hierarchy with self-constent source. Through inverse scattering transformation the exact solutions are got and a single soliton solution is exhibited explicitly as an example. The third one is a modified Ragnisco-Tu hierarchy. By applying inverse scattering transformation to both the isospectral and the nonisospectral hierarchy, the exact solutions to the hierarchy and the exact solutions to the correspondent hierarchy with self-constent source are found.Concretely, in chapter three, we investigate the equation deduced from Schrodinger spectral problem entirely. We get the inverse KdV hierarchy, the mixed KdV-inverse KdV hierarchy and inverse mixed KdV-MKdV hierarchy and relate the inverse KdV hierarchy to the associated Camassa-Holm hierarchy via a transformation. We discuss the bilineari-sation method of a hierarchy of evolution equation and succeed in bilinearizing the inverse KdV hierarchy through the recurrence relation of its Lax pair. In this chapter, we first prove the Darboux-Crum theorem of the classic KdV hierarchy and obtain its exact solutions. Then we prove a special result of the Darboux-Crum theorem to the inverse KdV hierarchy. Using the Backlund transformation provided by the theorem we present the new solutions especially the rational solutions producing from a nonzero seed solution. Then we pay our attention to a mixed KdV-inverse KdV equation and get its Wronskian solutions. Finally, we deduce the mixed KdV-MkdV equation through the generalized KdV operator and also get its Wronskian solutions. The method to bilinear Lax pair in this part develop the appli-cation of Wronskian technique.In chapter five, we deduce a Ragnisco-Tu hierarchy with self-constent source(RTHSS) and get the conservation law of Ragnisco-Tu equation firstly. For the nonlinear Lax-integrable equation with discretized space variable,IST play its role only when the spectral equation has asymptotic solution z±n. As far as this is concerned, some nonlinear Lax-integrable equations don’t meet the need. So a transformation of spectral function must be given to change the situation. For the RTHSS, we give a suitable spectral equation and prove the existence, differentiability and uniqueness of its Jost solutions. Further, we normalize the eigenfunction of spectral equation smoothly and solve the RTHSS via IST finally. At the end of this chapter, we choose to exhibit a single soliton solution of RTHSS explicitly.In chapter six, we generalize the spectral problem of Ragnisco-Tu equation and de-rive the isospectral and non-isospectral generalized Ragnisco-Tu hierarchies. For a special one(we call it as modified Ragnisco-Tu hierarchy(MRTH)), we study them using the IST simultaneously. The formula of exact solutions are obtained under the no reflection condi-tion. Further, we apply IST to the MRTH with self-constent source and also obtain its exact solutions. We find that the determination of normalization constant of RTHSS and MRTH has common point but they are slightly different from that for the Toda lattice, Ablowitz-Ladik lattice.