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

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 $N\to \mathrm{\infty }$, with the discrete spectrum confined to two segments ${\mathrm{\Sigma }}_1$ and ${\mathrm{\Sigma }}_2$. 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 $x\to -\mathrm{\infty }$, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime $t\to +\mathrm{\infty }$, we assume that the endpoint $F$ lies on the trajectory of $H(\xi )$ with $\xi =\frac{x}{2t}\in (-E_1-\sqrt{2}{E}_2,-E_1)$, namely, $F=H(\widehat{\xi })$, $\widehat{\xi }\in (-E_1-\sqrt{2}{E}_2,-E_1)$. Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable $\xi =\frac{x}{2t}$. More precisely, there exist an exponentially decaying region $\xi \in (-E_1,+\mathrm{\infty })$, a modulated elliptic-wave region $\xi \in (\widehat{\xi },-E_1)$, and an unmodulated elliptic-wave region $\xi \in (-\mathrm{\infty },\widehat{\xi })$.