Exponential asymptotics for translational modes in the discrete nonlinear Schrödinger 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 Schrödinger 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.