Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: exact solutions and stability

TL;DR

This paper constructs explicit higher-order rogue-wave solutions in a coupled Ablowitz-Ladik system, analyzes their stability, and demonstrates their near stability or instability through simulations, advancing understanding of discrete rogue waves.

Contribution

It introduces a discrete generalized Darboux transformation to derive new higher-order two-component rogue-wave solutions and studies their stability properties.

Findings

Higher-order discrete rogue-wave solutions are explicitly constructed.

Tightly bound RWs are nearly stable, loosely bound RWs are strongly unstable.

Systematic simulations confirm stability characteristics.

Abstract

An integrable system of two-component nonlinear Ablowitz-Ladik (AL) equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.