Rigorous Asymptotics of a KdV Soliton Gas

Abstract

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→ + ∞ of a gas of N-solitons. We show that this gas of solitons in the limit N→ ∞ is slowly approaching a cnoidal wave solution for x→ - ∞ up to terms of order O(1 / x) , while approaching zero exponentially fast for x→ + ∞. 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.