A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations

Abstract

By introducing a spectral problem with an arbitrary parameter, we derive a Kaup–Newell-type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the Kaup–Newell (KN) equation, the Chen–Lee–Liu (CLL) equation, the Gerdjikov–Ivanov (GI) equation, the Burgers equation, the modified Korteweg-deVries (MKdV) equation and the Sharma–Tasso–Olver equation. It is shown that the hierarchy is integrable in Liouville’s sense and possesses multi-Hamiltonian structure. Under the Bargann constraint between the potentials and the eigenfunctions, the spectral problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system. The involutive representation of the solutions for the Kaup–Newell-type hierarchy is also presented. In addition, an N-fold Darboux transformation of the Kundu equation is constructed with the help of its Lax pairs and a reduction technique. According to the Darboux transformation, the solutions of the Kundu equation is reduced to solving a linear algebraic system and two first-order ordinary differential equations. It is found that the KN, CLL, and GI equations can be described by a Kundu-type derivative nonlinear Schrödinger equation involving a parameter. And then, we can construct the Hamiltonian formulations, Lax pairs and N-fold Darboux transformations for the Kundu, KN, CLL, and GI equations in explicit and unified ways.