N-fold Darboux Transformation For The Integrable Couplings Of Kaup-Newell Equations And Its Solutions

In this thesis,we will construct the determinant representations of N-fold Darboux transformation(DT)and N-transformed solutions for the integrable couplings of Kaup-Newell(KN)equations.When the reduction conditions are considered,we obtain ones for the integrable couplings of derivative nonlinear Schrodinger(DNLS)equations.Starting from the zero seed solutions and non-zero seed solutions,we respectively obtain some particular solutions for the integrable couplings of DNLS equations.There are five parts in this thesis.In the first section,we will mainly recall some knowledge associated with this thesis,which includes the origin of soliton and integrable system,the development of DT,the construction of integrable couplings equations and so on.Naturally,we will propose the construction for the DT of the integrable couplings of KN equations.In Section 2,we will first review the construction of KN equations.Then through enlarging spectral problem associated with soliton equations(i.e.4 x 4 matrix instead of 2 x 2 matrix in the spatial spectral matrix),we will obtain the integrable couplings of KN equations by a usual method.Last,by making use of the trace identity,we can rewrite the integrable couplings of KN equations as the bi-Hamiltonian form.In Section 3,for the integrable couplings of KN equations,we will construct one-fold DT,and then by iterating once,we will derive two-fold DT.We find that there exists a significant difference between the Darboux matrix of the one-fold DT and one of the two-fold DT.Therefore,when N is an odd number and N is an even number,two types of DT should be discussed.Correspondingly,the determinant expressions of the N-transformed solutions p[N],q[N],r[N]and s[N]will be given.In Section 4,when the reduction conditions q=-p*and s=-r*are imposed on the integrable couplings of KN equations,we will obtain the integrable couplings of DNLS equations.So if we choose the reduced eigenvalue and eigenfunctions,we will derive determinant representations of the N-fold DT(N is an odd number and N is an even number)and N-transformed solutions for the integrable couplings of DNLS equations.In Section 5,on one hand,starting from the zero seed solutions,i.e.p=q=r=s=0,we obtain solutions of N=1 and N=2,respectively.On the other hand,starting from the non-zero periodic solutions,we also obtain solutions of N=1 and N=2,respectively.Unfortunately,the latter solutions are very tedious.So we don't list them in this thetis.