The Correspondence Between Integrable Hierarchies

Soliton theory plays an important role in natural science,and it has been widely used in many fields,such as mathematics?physics and biology.In re-cent years,the nonlinear integrable systems which admit solitons,such as Korteweg-de Vries(KdV)equation,Camassa-Holm(CH)equation,the nonlinear schrodinger equation?have been well studiedMany nonlinear integrable systems are not independent.They are related by the transformations between dependent and independent variables of the e-quations.For instance,the CH equation can be constructed by tri-Hamiltonian duality method from the bi-Hamiltonian structure of KdV equation,the KdV equation and modified Korteweg-de Vries(mKdV)equation are related through the Miura transformation,the KdV hierarchy can be connected with the CH hierarchy by a Liouville transformation.In this thesis,we study two explict cor-responfences between the short-wave model of Novikov hierarchy and Sawada-Kotera(SK)hierarchy,and between the integrable generalized schrodinger hier-archy and derivative schrodinger hierarchyFirstly,we show how the Liouville transformation between the isospectral problems of the short-wave model of Novikov and the SK equations,relates the corresponding hierarchies,in both positive and negative directions,as well as their associated Hamiltonian conservation laws.Next,we establish the corre-spondence between the generalized schrodinger and derivative schrodinger hier-archies,based on the gauge transformation between their respective isospectral problems.