Wigner instability analysis of the damped Hirota equation

TL;DR

This paper investigates the modulation instability of the damped Hirota equation using the Wigner function approach, revealing the persistence of baseband instability despite damping and the resilience of Kerr-induced structures.

Contribution

It provides a novel analysis of the Hirota equation's modulation instability under stochastic and damping effects, highlighting the stability characteristics of Kerr interactions and the impact of damping spectra.

Findings

Modulation instability remains baseband type despite damping.

Damping reduces the unstable spectrum regardless of higher-order effects.

Kerr-induced instability structures are resilient to third-order dispersion stabilization.

Abstract

We address the modulation instability of the Hirota equation in the presence of stochastic spatial incoherence and linear time-dependent amplification/attenuation processes via the Wigner function approach. We show that the modulation instability remains baseband type, though the damping mechanisms substantially reduce the unstable spectrum independent of the higher-order contributions (e.g. the higher-order nonlinear interaction and the third-order dispersion). Additionally, we find out that the unstable structure due to the Kerr interaction exhibits a significant resilience to the third-order-dispersion stabilizing effects in comparison with the higher-order nonlinearity, as well as a moderate Lorentzian spectrum damping may assist the rising of instability. Finally, we also discuss the relevance of our results in the context of current experiments exploring extreme wave events driven by the modulation instability (e.g. the generation of the so-called rogue waves).