Rogue waves arising on the standing periodic waves in the Ablowitz-Ladik equation

TL;DR

This paper investigates rogue waves on standing periodic solutions of the Ablowitz-Ladik equation, relating stability spectra to Lax spectra, and introduces a novel non-standard Lax system to analyze rogue wave formation.

Contribution

It introduces a new non-standard Lax system for the Ablowitz-Ladik equation and derives analytical expressions for rogue waves on standing periodic backgrounds.

Findings

Spectral bands and eigenfunctions characterize rogue wave emergence.

Magnification factors of rogue waves are analytically computed.

Comparison with continuous models highlights differences in rogue wave behavior.

Abstract

We study the standing periodic waves in the semi-discrete integrable system modelled by the Ablowitz-Ladik equation. We have related the stability spectrum to the Lax spectrum by separating the variables and by finding the characteristic polynomial for the standing periodic waves. We have also obtained rogue waves on the background of the modulationally unstable standing periodic waves by using the end points of spectral bands and the corresponding eigenfunctions. The magnification factors for the rogue waves have been computed analytically and compared with their continuous counterparts. The main novelty of this work is that we explore a non-standard linear Lax system, which is different from the standard Lax representation of the Ablowitz-Ladik equation.