The Ablowitz-Ladik equation featuring two time-varying coefficients: Continuum limit, modulation instability, novel localized wave solutions and related dynamics

Abstract

This letter addresses the Ablowitz-Ladik equation featuring two time-varying coefficients. Initially, we investigate its continuum limit and modulation instability. Subsequently, we extend the generalized $(p,N-p)$-fold Darboux transformation to this variable-coefficient equation, from which we derive a variety of exact solutions such as breather, rogue wave, periodic wave, as well as their combined interaction solutions with two time-varying coefficients on diverse seed backgrounds, and their localized wave structures are analyzed and discussed graphically. Intriguingly, due to the effects of variable coefficients and control parameters, we discover that a fundamental rogue wave characterized by a single peak and two valleys can be divided into two fundamental rogue waves, and we further obtain a novel rogue wave situated on a kink soliton background, while these new phenomena cannot be found in the constant coefficient Ablowitz Ladik equation. These innovative discoveries will broaden our understanding of particular physical phenomena in optical fiber.