Continuum limits for rational, breather solutions, and gauge equivalent model of the generalized discrete nonlinear Schrödinger equation

Abstract

In this paper, we focus on an integrable higher‐order discrete nonlinear Schrödinger (dNLS) equation, which yields the Lakshmanan–Porsezian–Daniel (LPD) equation in the continuum limit. The breather solutions of the higher‐order dNLS are obtained for the first time. We show that the rational solutions and breather solutions of the higher‐order dNLS equation yield the counterparts of the integrable fourth‐order NLS equation under proper continuous limits. Most importantly, we also explore the gauge equivalent equation of the higher‐order dNLS equation from the point of view of the continuous limit theory. We make integrable discretization for the linear spectrum problem of the integrable generalized Heisenberg ferromagnetic equation that is just the gauge equivalent structure of the LPD equation. Compatibility condition of the discrete linear spectrum problem leads to a discrete integrable generalized Heisenberg Ferromagnetic equation. It is revealed that the discrete linear spectrum problem of the discrete integrable generalized ferromagnetic equation converges to the corresponding spectrum problem of the integrable generalized Feisenberg spin model in the continuum limit.In addition, with the help of the method of linear combinations for the conserved fluxes and densities, the conservation laws of the integrable higher‐order dNLS equation converge to the homologous results of the LPD equation.