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

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.