Standing lattice solitons in the discrete NLS equation with saturation

TL;DR

This paper investigates standing lattice solitons in the discrete nonlinear Schrödinger equation with saturation, analyzing singularities in complex plane to explain transparent points where the Peierls-Nabarro barrier vanishes.

Contribution

It provides a detailed analysis of singularities in the complex plane of solitary waves, explaining transparent points in the discrete NLSS and its continuum approximations.

Findings

Existence of quadruplet of logarithmic singularities near the real axis.

Identification of transparent points where the Peierls-Nabarro barrier vanishes.

Application of singularity analysis to continuum and delay differential equation models.

Abstract

We consider standing lattice solitons for discrete nonlinear Schrodinger equation with saturation (NLSS), where so-called transparent points were recently discovered. These transparent points are the values of the governing parameter (e.g., the lattice spacing) for which the Peierls-Nabarro barrier vanishes. In order to explain the existence of transparent points, we study a solitary wave solution in the continuous NLSS and analyse the singularities of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic singularities nearest to the real axis is proven and applied to two settings: (i) the fourth-order differential equation arising as the next-order continuum approximation of the discrete NLSS and (ii) the advance-delay version of the discrete NLSS.