Dark solitons in the fractional NLS equation

TL;DR

This paper investigates dark solitons in the fractional nonlinear Schrödinger equation, demonstrating their existence, stability properties, and complex dynamics including breathing behaviors and potential reduced models.

Contribution

It presents the first analysis of dark fractional solitary waves, their stability, two-soliton solutions, and associated dynamics, including novel breathing orbits and ODE interaction models.

Findings

Dark solitary waves exist and are stable in fractional NLS.

Two-soliton solutions exhibit multiple branches with different stability properties.

Unprecedented breathing dynamics and periodic orbits are observed.

Abstract

In the present work we consider the subject of dark fractional solitary waves in the realm of generalized (fractional) forms of the nonlinear Schr\"odinger (NLS) equation. While earlier studies have examined such states in the realm of real field theories, we showcase the existence and stability of individual dark solitary waves in such NLS settings and subsequently turn to two-soliton solutions. We find different branches of such two-soliton solution equilibria and contrary to the real field-theoretic setting all possible branches of two-soliton equilibria are found to be potentially unstable, although with different types of instabilities. Odd branches are potentially subject to oscillatory instabilities, while even branches are always exponentially unstable. The dynamics that results from the instabilities is also examined and is found to potentially feature breathing characteristics. This prompts us to seek and find associated periodic (breathing) orbits that are also unprecedented in this context, to the best of our knowledge. The effective particle-like dynamics of the solitary waves also prompts us to seek ordinary differential equation (ODE) descriptions to the dark soliton interaction dynamics. These are shown to hold promise toward providing us with effective reduced order models, indicating some potential directions for further investigation.