Long-time Asymptotics of a Full Camassa-Holm Soliton Gas

We investigate the long-time asymptotics of a full soliton gas for the Camassa--Holm equation. The analysis starts from a pure-soliton Riemann--Hilbert (RH) problem with \(2N\) poles and two distinct types of residue conditions. We prove that, as \(N\to\infty\), this discrete RH problem converges to a limiting soliton gas RH problem whose jump matrix contains two nonzero reflection coefficients. In this sense, the limiting problem gives a full soliton gas model for the Camassa--Holm equation, in contrast to the previously studied half soliton gas models, whose jump matrices involve only one nonzero reflection coefficient. The limiting RH problem is analyzed by the Deift--Zhou nonlinear steepest descent method. The presence of two nonzero reflection coefficients requires two different types of triangular factorizations of the jump matrix and leads to a more delicate \(g\)-function mechanism. The main difficulty lies in the construction of suitable \(g\)-functions adapted to the Camassa--Holm phase, together with the precise control of their behavior near the distinguished point \(k=i/2\) and at infinity. Depending on the location of the spectral endpoints \(\eta_1\) and \(\eta_2\), different \(g\)-function mechanisms arise. In this paper, we focus on Case I and derive the long-time asymptotic formulas in three elliptic-wave regions of the self-similar plane. In each region, the leading term is given by a finite-gap elliptic function, while in the central region the first correction is of order \(\mathcal O(t^{-1/2})\) and involves parabolic cylinder functions.