Darboux Transformations Of Some Integrable Systems

This thesis presents a unified approach to study Darboux-B(?)cklund transformations of some Camassa-Holm(CH)type equations.This method simplifies the approach presented by Qiao et al.to obtain soliton solutions of the CH equation by the Darboux transformation,because it does not need to use the value of the wave function of the spectral problem of the CH type equation when the spectral parameter is zero(the spectral parameter is zero,which is the singular point of the time part of Lax pair),and avoids computing asymptotic properties of the wave functions.Then we apply this method to some CH type equations and a new system.This dissertation is divided into four chapters as follows.In chapter one,we introduce the research background,main results and some knowledge needed in the research.In chapter two,we propose a unified method to solve the Darboux-B(?)cklund transformation of the CH type equations,and apply it to the CH,the m CH,the 2-CH,the Degasperis-Procesi and the Novikov equations,which yield Darboux-B(?)cklund transformations of these equations.Especially,we present three new Darboux-B(?)cklund transformations of the 2-CH system.In chapter three,a new system related to the first negative flow in AKNS hierarchy is studied by the above method,we construct three Darboux-B(?)cklund transformations for the new system and obtain some soliton solutions.In chapter four,we give discussion and conclusion of this thesis.