Borel-Pad\'e exponential asymptotics for the discrete nonlinear Schr\"odinger model with next-to-nearest neighbour interactions

TL;DR

This paper develops Borel-Padé exponential asymptotics to analyze the stability eigenvalues of stationary states in a discrete nonlinear Schrödinger model with competing next-nearest neighbor interactions, revealing exponential dependence on coupling.

Contribution

It introduces a novel Borel-Padé exponential asymptotics approach to compute hidden Stokes multipliers in discrete nonlinear Schrödinger models with complex interactions.

Findings

Eigenvalues depend exponentially on coupling parameter  with polynomial corrections.

Good agreement between asymptotic predictions and numerical computations.

Method reveals subdominant Stokes multipliers beyond all asymptotic orders.

Abstract

In the present work we study discrete nonlinear Schr{\"o}dinger models combining nearest (NN) and next-nearest (NNN) neighbor interactions, motivated by experiments in waveguide arrays. While we consider the more experimentally accessible case of positive ratio $\mu$ of NNN to NN interactions, we focus on the intriguing case of competing such interactions $(\mu<0)$, where stationary states can exist only for $-1/4 < \mu < 0$. We analyze the key eigenvalues for the stability of the pulse-like stationary (ground) states, and find that such modes depend exponentially on the coupling parameter $\eps$, with suitable polynomial prefactors and corrections that we analyze in detail. Very good agreement of the resulting predictions is found with systematic numerical computations of the associated eigenvalues. This analysis uses Borel-Pad\'{e} exponential asymptotics to determine Stokes multipliers in the solution; these multipliers cannot be obtained using standard matched asymptotic expansion approaches as they are hidden beyond all asymptotic orders, even near singular points. By using Borel-Pad\'{e} methods near the singularity, we construct a general asymptotic template for studying parametric problems which require the calculation of subdominant Stokes multipliers.