Multi-soliton, breather and rogue wave solutions of a generalized Kaup-Newell system

In this paper, we derive a generalized Kaup-Newell system corresponding to a 4 × 4 matrix spectral problem by means of the zero-curvature equation, and construct the bi-Hamiltonian structures of this system. Significantly, a new coupled derivative nonlinear Schrödinger equation is produced from the generalized Kaup-Newell system. Moreover, a Darboux transformation of the generalized system is investigated, and several types of multi-soliton solutions, including single-hump and double-hump solitons, are obtained. As a specific reduction, the Darboux transformation of the coupled derivative nonlinear Schrödinger equation is established, and its soliton, breather on an oscillating background, and rogue wave solutions are derived. These results are helpful to understand the integrable properties and dynamic behaviors of integrable equations related to higher-order spectral problems.