Exponential Asymptotics for Translational Modes in the Discrete Nonlinear Schr{\"o}dinger Model

TL;DR

This paper applies exponential asymptotics to analyze translational eigenmodes in the discrete nonlinear Schrödinger equation, providing precise scaling laws and corrections, and offering a general approach for similar lattice models.

Contribution

It introduces a detailed exponential asymptotics methodology for translational modes, including higher-order corrections, in discrete nonlinear lattice models.

Findings

Confirmed exponential scaling of eigenvalues as e^{-π^2/(2ε)}

Derived a power-law prefactor of ε^{-5/2}

Provided the first accurate leading-order prefactor and correction

Abstract

In the present work, we revisit the topic of translational eigenmodes in discrete models. We focus on the prototypical example of the discrete nonlinear Schr{\"o}dinger equation, although the methodology presented is quite general. We tackle the relevant discrete system based on exponential asymptotics and start by deducing the well-known (and fairly generic) feature of the existence of two types of fixed points, namely site-centered and inter-site-centered. Then, turning to the stability problem, we not only retrieve the exponential scaling (as \( e^{-\pi^2/(2 \varepsilon)} \), where \( \varepsilon \) denotes the spacing between nodes) and its corresponding prefactor power-law (as \( \varepsilon^{-5/2} \)), both of which had been previously obtained, but we also obtain a highly accurate leading-order prefactor and, importantly, the next-order correction, for the first time, to the best of our knowledge. This methodology paves the way for such an analysis in a wide range of lattice nonlinear dynamical equation models.