Long-time asymptotics of a full arbitrary-genus dark soliton gas for the defocusing nonlinear Schrodinger equation

Abstract

We introduce a full arbitrary-genus dark soliton gas for the defocusing nonlinear Schr\"odinger equation with finite-density boundary conditions. Starting from a generalized meromorphic Riemann--Hilbert problem with two alternating residue families on each unit-circle arc, we derive an exact thermodynamic limit whose jump matrix contains two nonzero continuum densities. The limiting Riemann--Hilbert problem is uniquely solvable. In contrast with the half dark-soliton gas, every spectral arc of the full gas carries both oscillatory exponentials. We analyze the resulting problem by the Deift--Zhou nonlinear steepest-descent method on a fixed genus-$N$ spectral curve. The moving point in each mixed sector is a stationary factorization-switching point, not a branch point. The active arc is split into two parts and opened crosswise, while lenses are opened around every remaining arc. After removal of exponentially small lens jumps, the model contour therefore retains all $N$ spectral arcs in every self-similar sector. A quotient-curve zero-counting argument proves strict monotonicity of the characteristic velocity and the global ordering of all endpoint velocities, so the self-similar line is divided into $2N+1$ nonempty sectors. The leading term is an $N$-dimensional Riemann-theta finite-gap solution. The error is $O(t^{-1})$ in the $N+1$ pure sectors and $O(t^{-1/2})$ in the $N$ mixed sectors, uniformly away from the critical rays.