Wave Breaking and Structure-Preserving Fully Discrete Method for Stochastic Camassa--Holm Equation with Additive Noise
Abstract
In this article, we investigate the stochastic Camassa--Holm equation driven by additive noise, which models nonlinear shallow-water waves under random external forcing. We establish that, almost surely, any finite-time breakdown occurs through wave breaking: the wave amplitude and the positive part of the slope remain bounded, whereas the minimum slope tends to −∞. Under an additional moment assumption, we derive an initial-slope condition ensuring wave breaking with positive probability. We also formulate the equation as a stochastic Hamiltonian PDE, establish its stochastic multi-symplectic conservation law, and derive the evolution law for the averaged H1 energy. Combining the Fourier--Galerkin method, symplectic Runge--Kutta integration, and exact solution of the linear stochastic subsystem, we propose a fully discrete method which preserves an integrated discrete stochastic multi-symplectic conservation law and the linear growth of the discrete averaged H1 energy. Under suitable regularity and resolution assumptions, we present that the fully discrete dynamics retain the slope-steepening mechanism responsible for wave breaking, providing a discrete counterpart of the continuous behavior.
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BibTeX
@misc{sun2026wave,
title = {{Wave Breaking and Structure-Preserving Fully Discrete Method for Stochastic Camassa--Holm Equation with Additive Noise}},
author = {Liying Sun and Zizhao Sun and Liying Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32128},
eprint = {2609.32128},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32128},
}
RIS
TY - UNPB
TI - Wave Breaking and Structure-Preserving Fully Discrete Method for Stochastic Camassa--Holm Equation with Additive Noise
AU - Liying Sun
AU - Zizhao Sun
AU - Liying Zhang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32128
UR - https://arxiv.org/abs/2609.32128
AB - In this article, we investigate the stochastic Camassa--Holm equation driven by additive noise, which models nonlinear shallow-water waves under random external forcing. We establish that, almost surely, any finite-time breakdown occurs through wave breaking: the wave amplitude and the positive part of the slope remain bounded, whereas the minimum slope tends to −∞. Under an additional moment assumption, we derive an initial-slope condition ensuring wave breaking with positive probability. We also formulate the equation as a stochastic Hamiltonian PDE, establish its stochastic multi-symplectic conservation law, and derive the evolution law for the averaged H^1 energy. Combining the Fourier--Galerkin method, symplectic Runge--Kutta integration, and exact solution of the linear stochastic subsystem, we propose a fully discrete method which preserves an integrated discrete stochastic multi-symplectic conservation law and the linear growth of the discrete averaged H^1 energy. Under suitable regularity and resolution assumptions, we present that the fully discrete dynamics retain the slope-steepening mechanism responsible for wave breaking, providing a discrete counterpart of the continuous behavior.
ER -