We analyze a new class of solutions of the modified KdV equation introduced by Dyachenko, Zakharov, and Zakharov, and which can be interpreted as a soliton gas. These solutions are characterized by a Riemann-Hilbert problem which arises as the limit $N\to \infty$ of a sequence of $2N$-soliton RH problems, where the solitons display two different triangularity in the jumps. The limiting RH problem belongs to a novel class of problems, where the jump matrices do not have any zero entry (as opposed to the classical upper/lower triangular setting).
We show that the initial profile of such a gas solution is approaching a cnoidal wave solution for $x\to \pm \infty$ with different phase shifts, and 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, with slowly modulated phase shifts as time varies from $-\infty$ to $+\infty$.
This is a joint work with Ken McLaughlin (Tulane U.) and Robert Jenkins (UCF).

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