The Complex Hamiltonian Systems And Quasi-periodic Solutions In The Derivative Nonlinear Schr(?)dinger Equations

The purpose of this work is to apply complex FDHSs for a construction of quasi-periodic solution of the DNLS hierarchy.In order to simplify the procedure for getting explict solutions,the DNLS hierarchy is reduced to a family of complex FDHSs by using the nonlinearization of Lax pair.From the Lax representations of DNLS hierarchy,we get a Lax matrix whose determinant gives rise to integrals of motion and a hyperelliptic curve for complex FDHSs,we also prove this matrix satisifies a Lax equation.With a set of quasi-Abel-Jacobi variables,the Liouville integrability of complex FDHSs is proved.Moreover,involutive solutions of the complex FDHSs are shown to be finite parametric solutions of the DNLS hierarchy,and the used Bargmann map specifies a finite-dimensional invariant subspace to the DNLS flows.Suitably choose the Abel-Jacobi(or angle)variable to straighten out the DNLS flows,the evolution behavior of flows on the Jacobi variety of an invariant Riemann surface could be displayed.In the end,resorting to the Riemann theorem,we get quasi-periodic solutions of DNLS hierarchy by applying the Riemann-Jacobi inversion to the Abel-Jacobi solutions of DNLS flows.