On the proximity of Ablowitz-Ladik and discrete Nonlinear Schr\"odinger models: A theoretical and numerical study of Kuznetsov-Ma solutions

TL;DR

This study explores the relationship between Ablowitz-Ladik and discrete nonlinear Schrödinger models, demonstrating that Kuznetsov-Ma breather solutions are closely approximated in the non-integrable DNLS model, with implications for rogue wave dynamics.

Contribution

It provides a theoretical and numerical comparison of KM solutions in integrable and non-integrable lattice models, including bounds on solution proximity and bifurcation analysis.

Findings

Proximity of KM solutions between AL and DNLS models for certain parameters.

Derived and numerically confirmed bounds on solution differences.

Identified parameter regimes for stable KM-type breathers in DNLS.

Abstract

In this work, we investigate the formation of time-periodic solutions with a non-zero background that emulate rogue waves, known as Kuzentsov-Ma (KM) breathers, in physically relevant lattice nonlinear dynamical systems. Starting from the completely integrable Ablowitz-Ladik (AL) model, we demonstrate that the evolution of KM initial data is proximal to that of the non-integrable discrete Nonlinear Schr\"odinger (DNLS) equation for certain parameter values of the background amplitude and breather frequency. This finding prompts us to investigate the distance (in certain norms) between the evolved solutions of both models, for which we rigorously derive and numerically confirm an upper bound. Finally, our studies are complemented by a two-parameter (background amplitude and frequency) bifurcation analysis of numerically exact, KM-type breather solutions to the DNLS equation. Alongside the stability analysis of these waveforms reported herein, this work additionally showcases potential parameter regimes where such waveforms with a flat background may emerge in the DNLS setting.