Long-time asymptotics of a full Camassa-Holm soliton gas
Abstract
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→∞, 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 η1 and η2, 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 𝒪(t−1/2) and involves parabolic cylinder functions.
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BibTeX
@misc{yan2026longtime,
title = {{Long-time asymptotics of a full Camassa-Holm soliton gas}},
author = {Dedi Yan and Xianguo Geng and Minxin Jia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.10727},
eprint = {2606.10727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.10727},
}
RIS
TY - UNPB
TI - Long-time asymptotics of a full Camassa-Holm soliton gas
AU - Dedi Yan
AU - Xianguo Geng
AU - Minxin Jia
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2606.10727
UR - https://arxiv.org/abs/2606.10727
AB - 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→∞, 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 η_1 and η_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 𝒪(t^−1/2) and involves parabolic cylinder functions.
ER -