This paper studies how a dense collection of bright solitons, called a soliton gas, behaves far away in space and time for the focusing nonlinear Schrödinger equation. The authors show that, under precise spectral assumptions, the wave field separates into three self-similar regions as time grows: one where the field decays exponentially, one where it looks like a slowly changing elliptic (periodic) wave, and one where it is an elliptic wave with fixed coefficients.

To build the soliton gas the researchers start from pure N-soliton solutions and let the number N go to infinity. In that limit the discrete spectral points that define the solitons condense onto two short segments in the complex plane. The paper works with this “continuum limit” and with pure-soliton initial data (no continuous reflection). The discrete data are taken to be equally spaced on the two segments and to discretize a smooth weight function along those segments.

The analysis uses the inverse-scattering formulation of the equation, cast as a matrix Riemann–Hilbert problem, and then applies the nonlinear steepest descent method together with a tailored g-function construction. The g-function is an analytic change of phase that turns rapidly oscillatory terms into constant jumps. A key technical step is proving the solvability of a nonlinear system that fixes the g-function parameters. Where that system is solvable, the remaining model problem can be solved in terms of theta functions on a genus-one Riemann surface, which gives the elliptic-wave formulas.

The main asymptotic conclusions are concrete. For fixed time and x→+∞ the soliton gas decays exponentially to zero. For x→−∞ it approaches a finite-gap elliptic solution with an explicit theta-function form up to an error of order 1/|x|. For large time t→+∞ with the self-similar variable ξ = x/(2t) fixed, the solution falls into three regions determined by ξ: an exponentially decaying region for ξ>−E1, a modulated elliptic-wave region for ξ between a special value ξ̂ and −E1, and an unmodulated elliptic-wave region for ξ<ξ̂. The special value ξ̂ is tied to the spectral endpoint F by the condition F = H(ξ̂). In the modulated region the theta-function solution has slowly varying parameters and the authors give asymptotic formulas with error terms of order 1/t.