Rigorous Asymptotics of a KdV Soliton Gas

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