Explicit solutions and Darboux transformations of a generalized D-Kaup–Newell hierarchy

Abstract

Darboux and Backlund transformations on integrable couplings are formulated and generalized. The generalization of the theory pertains to spectral problems where the spatial spectral matrix is a polynomial in $$\lambda $$ of any order. A specific application to a generalized D-Kaup–Newell integrable couplings system is worked out, along with an explicit formula for the associated Backlund transformation. Solutions are given for the 0, 1, 2, 3-order generalized D-Kaup–Newell integrable coupling system. Formulas for the general m-th-order integrable couplings system are seen. Graphs of explicit solutions to the fourth-order integrable couplings are presented for chosen parameters showing solitons. A brief discussion about open problems and physical implications of the paper concludes the paper.