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arXiv · 2609.32128

Wave Breaking and Structure-Preserving Fully Discrete Method for Stochastic Camassa--Holm Equation with Additive Noise

Liying Sun·Zizhao Sun·Liying Zhang
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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 $-\infty$. 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.

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Liying Sun, Zizhao Sun, Liying Zhang. 2026-09-26. Wave Breaking and Structure-Preserving Fully Discrete Method for Stochastic Camassa--Holm Equation with Additive Noise. https://arxiv.org/abs/2609.32128

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