Suppression of soliton collapses, modulational instability, and rogue-wave excitation in two-L\'evy-index fractional Kerr media

TL;DR

This paper investigates the dynamics of solitons, modulational instability, and rogue waves in a two-Lévy-index fractional Kerr medium, revealing suppression of collapse and conditions for rogue wave formation.

Contribution

It introduces a novel two-Lévy-index fractional nonlinear Schrödinger model and demonstrates soliton stabilization, instability analysis, and rogue wave generation within this framework.

Findings

Soliton collapse can be suppressed in the two-Lévy-index system.

Modulation instability occurs even in defocusing nonlinearity with opposite diffraction signs.

First- and second-order rogue waves are generated based on instability analysis.

Abstract

s in laser systems with two fractional-dispersion/diffraction terms, quantified by their L\'{e}vy indices, $\alpha_{1}\, \alpha_{2}\in (1, 2]$, and self-focusing or defocusing Kerr nonlinearity. Some fundamental solitons are obtained by means of the variational approximation, which are verified by comparison with numerical results. We find that the soliton collapse, exhibited by the one-dimensional cubic fractional nonlinear Schr\"{o}dinger equation with only one L\'{e}vy index $\alpha =1$, can be suppressed in the two-L\'{e}vy-index fractional nonlinear Schr\"{o}dinger system. Stability of the solitons is also explored against collisions with Gaussian pulses and adiabatic variation of the system parameters. Modulation instability of continuous waves is investigated in the two-L\'{e}vy-index system too. In particular, the modulation instability may occur in the case of the defocusing nonlinearity when two diffraction coefficients have opposite signs. Using results for the modulation instability, we produce first- and second-order rogue waves on top of continuous waves, for both signs of the Kerr nonlinearity.