Exponential Asymptotics for Dark Solitons of the Discrete NLS Model

TL;DR

This paper develops precise exponential asymptotics for pinned dark solitons in the discrete nonlinear Schrödinger equation, resolving previous discrepancies with numerical results and connecting to continuum theory.

Contribution

It provides the first accurate exponential asymptotics for discrete dark solitons and reconciles these with continuum models in external potentials.

Findings

Derived exact exponential asymptotics for pinned dark solitary waves.

Resolved previous inconsistencies between asymptotic analysis and numerical computations.

Connected discrete soliton results with continuum nonlinear Schrödinger theory.

Abstract

In the present work we revisit the problem of the dark solitary wave pinned in the discrete nonlinear Schr{\"o}dinger equation. In a number of recent studies, the methodology of exponential asymptotics was attempted to be utilized in this problem, however the results were not found to be fully in agreement with associated multiprecision numerical computations. Here we resolve this conundrum by finding precise exponential asymptotics for the pinned dark solitary waves. Moreover, we reconcile the relevant result with a general theory of pinned dark solitary waves in the {\it continuum} nonlinear Schr{\"o}dinger equations in the presence of external potentials.