Borel–Padé Exponential Asymptotics for the Discrete Nonlinear Schrödinger Model with Next-to-Nearest Neighbor Interactions

Abstract.

In the present work we study discrete nonlinear Schrödinger models combining nearest neighbor (NN) and next-nearest neighbor (NNN) interactions, motivated by experiments in waveguide arrays. We consider both the more experimentally accessible case of positive ratio [Formula: see text] of NNN to NN interactions, as well as the intriguing case of competing such interactions [Formula: see text], where stationary states can exist only for [Formula: see text]. 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 [Formula: see text], 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–Padé 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–Padé methods near the singularity, we construct a general asymptotic template for studying parametric problems which require the calculation of subdominant Stokes multipliers.