Large-space and Large-time Asymptotics for the Focusing Nonlinear Schrödinger Soliton Gas

Abstract

We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schrödinger equation. The soliton gas is constructed as the continuum limit of pure $N$-soliton solutions as $N\to\infty$, with the discrete spectrum confined to two segments $Σ_1$ and $Σ_2$. In particular, our framework does not require the discrete spectrum to be confined to the imaginary axis. By combining the nonlinear steepest descent method with an appropriate $g$-function mechanism, we show that, as $x\to-\infty$, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime $t\to+\infty$, we assume that the endpoint $F$ lies on the trajectory of $H(ξ)$ with $ξ=\frac{x}{2t}\in(-E_1-\sqrt{2}E_2,-E_1)$, namely, $F=H(\hatξ)$, $\hatξ\in (-E_1-\sqrt{2}E_2,-E_1)$. Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable $ξ=\frac{x}{2t}$. More precisely, there exist an exponentially decaying region $ξ\in(-E_1,+\infty)$, a modulated elliptic-wave region $ξ\in(\hatξ,-E_1)$, and an unmodulated elliptic-wave region $ξ\in(-\infty,\hatξ)$.