Integrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation

Abstract

A spectral problem and the associated Gerdjikov–Ivanov (GI) hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known GI equation of derivative nonlinear Schrödinger equations is obtained. It is shown that the GI hierarchy is integrable in a Liouville sense and possesses bi-Hamiltonian structure. Moreover, the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenfunctions. In particular, an explicit N-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique. Some explicit solitonlike solutions of the GI equation are given by applying its Darboux transformation.