Rigorous Asymptotics of a KdV Soliton Gas

We analytically study the long time and large space asymptotics of a new broad class solutions KdV equation introduced by Dyachenko, Zakharov, Zakharov. These are characterized Riemann--Hilbert problem which we show arises as limit $N\to \infty$ gas $N$-solitons. that this solitons in $N \to is slowly approaching cnoidal wave solution for $x - (up to terms order $\mathcal{O} (1/x)$), while zero exponentially fast $x\to+\infty$. establish an asymptotic description times valid over entire spatial domain, Jacobi elliptic functions.