Long‐time asymptotics of a full soliton gas for the Camassa–Holm equation

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 poles and two distinct types of residue conditions. We prove that, as , 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 ‐function mechanism. The main difficulty lies in the construction of suitable ‐functions adapted to the Camassa–Holm phase, together with the precise control of their behavior near the distinguished point and at infinity. Depending on the location of the spectral endpoints and , different ‐function mechanisms arise. In this paper, we focus on Case I and derive the long‐time asymptotic formulae 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 and involves parabolic cylinder functions.