Exponential Asymptotics for Dark Solitons of the Discrete NLS Model

Abstract: In the present work we revisit the problem of the dark solitary wave pinned in the discrete nonlinear Schr{ö}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{ö}dinger equations in the presence of external potentials.