Abstract
In this paper, we investigate the smooth positon and breather-positon solutions of the generalized integrable discrete nonlinear Schrödinger (NLS) equation by the degenerate 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 space-time 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 the coefficient of nonlinear term on these discrete smooth positon solutions and breather-positon 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 has an opposite effect in two waveforms.
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