Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz-Ladik lattices

TL;DR

This paper investigates modulation instability and wave recurrence in Ablowitz-Ladik lattices, revealing instability conditions, constructing exact solutions, and describing evolution leading to singularities or recurrence, with perfect numerical agreement.

Contribution

It provides a detailed analysis of modulation instability in Ablowitz-Ladik lattices, constructs exact solutions via Darboux transformations, and explains the evolution and singularity formation of solutions.

Findings

Background instability for $AL_{-}$ when amplitude > 1.

Exact periodic solutions describing instability are constructed.

Solutions of $AL_{-}$ become singular in finite time for large amplitude.

Abstract

The Ablowitz-Ladik equations, hereafter called $AL_+$ and $AL_-$, are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schr\"odinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of $AL_{\pm}$ in the periodic setting, showing in particular that the background solution of $AL_{-}$ is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than $1$, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of $AL_{\pm}$ describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of $AL_-$ are always singular on curves of spacetime. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution i) evolves according to $AL_{+}$ into a recurrence of the above exact solutions, in the case of one and two unstable modes, and ii) evolves according to $AL_{-}$ into a singularity in finite time if the amplitude of the background is greater than $1$. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect.