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