Arbitrary-genus dark soliton gases in the defocusing nonlinear Schr\"{o}dinger hydrodynamics

TL;DR

This paper analytically studies the large-scale behavior and evolution of dark soliton gases in defocusing nonlinear Schrödinger hydrodynamics, revealing complex genus transitions and embedding phenomena.

Contribution

It introduces an arbitrary-genus dark soliton gas model and analyzes its asymptotics and evolution using the Deift-Zhou steepest descent method, extending understanding of soliton interactions.

Findings

The genus-$N$ soliton gas approaches finite-gap solutions at $x o - $ and background at $x o + $.

Long-time evolution shows a cascade from higher to lower genus regions.

Lower-genus gases' dynamics are embedded within higher-genus gases, governed by spectral properties.

Abstract

The defocusing nonlinear Schr\"{o}dinger hydrodynamics supports exact dark solitons under finite density boundary conditions. However, the dark soliton gas, an interacting ensemble of dark solitons, has not yet been studied. In this work, we introduce an arbitrary-genus potential of dark soliton gases by considering the limit of the $\mathcal{N}$-dark soliton as $\mathcal{N}\to \infty$. The large-space asymptotics and long-time evolution of this dark soliton gas potential are analytically investigated through Deift-Zhou nonlinear steepest descent approach. The genus-$N$ dark soliton gas potential approaches the genus-$N$ finite-gap solution as $x \to -\infty$ and the background $1$ as $x \to +\infty$. In the long-time evolution, as the self-similar variable $\xi=x/t$ increases, the gas configuration exhibits a cascade of behaviours, passing from unmodulated and modulated genus-$N$ regions and progressively reducing the genus down to the planar region (unmodulated genus-$0$ region). Notably, the evolution of lower-genus soliton gases can be embedded within that of higher-genus gases, exhibiting identical dynamics within specific regimes. This phenomenon is encoded by the underlying spectra. We also include numerical validations, in perfect agreement with the theoretical predictions.