Nth-order smooth positon and breather-positon solutions for the generalized integrable discrete nonlinear Schrödinger equation

Abstract

In this paper, we investigate the smooth positon and breather-positon solutions of the a generalized integrable discrete nonlinear Schrödinger (NLS) equation by the generalized Darboux transformation (DT). Starting from the zero seed solution, the Nth-order smooth positon solutions are obtained by degenerate DT. The breather solutions including Akhmediev breather, Kuznetsov-Ma breather and spacetime periodic breather are derived from the nonzero seed solution. Then the breather-positon solutions are constructed by gradual Taylor series expansion of the eigenfunctions in breather solutions. We study the effect of higher-order nonlinear terms on these discrete smooth positon solutions and breatherpositon solutions, which demonstrates that the interacting region of soliton-positon and breather-positon are highly compressed by higher-order nonlinear effects, but the distance between the two positons have an opposite effect in two waveforms.