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

C. J. Lustri; P. G. Kevrekidis; S. J. Chapman

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arXiv · 2501.08534

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

C. J. Lustri·P. G. Kevrekidis·S. J. Chapman

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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.

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C. J. Lustri, P. G. Kevrekidis, S. J. Chapman. 2025-01-15. Exponential Asymptotics for Translational Modes in the Discrete Nonlinear Schr{\"o}dinger Model. https://arxiv.org/abs/2501.08534

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