发表机构
Academy for Multidisciplinary Studies, Capital Normal University; School of Mathematical Science, China University of Mining and Technology (Beijing)（首都师范大学多学科研究院; 中国矿业大学（北京）数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带加性噪声的随机Camassa--Holm方程，证明有限时间破裂几乎必然通过波浪破碎发生，并构造保持随机多辛守恒律及平均H^1能量线性增长的全离散方法，且离散动力学保留斜率陡化机制。
AI 中文摘要
本文研究了由加性噪声驱动的随机Camassa--Holm方程，该方程描述了随机外力作用下非线性浅水波的传播。我们证明了，几乎必然地，任何有限时间内的破裂都是通过波浪破碎发生的：波幅和斜率的正部保持有界，而最小斜率趋于$-\infty$。在额外的矩假设下，我们导出了一个初始斜率条件，确保波浪破碎以正概率发生。我们还将该方程表述为随机Hamiltonian偏微分方程，建立了其随机多辛守恒律，并推导了平均$H^1$能量的演化规律。结合Fourier--Galerkin方法、辛Runge--Kutta积分以及线性随机子系统的精确解，我们提出了一种全离散方法，该方法保持了一个积分形式的离散随机多辛守恒律以及离散平均$H^1$能量的线性增长。在适当的正则性和分辨率假设下，我们证明了全离散动力学保留了导致波浪破碎的斜率陡化机制，为连续行为提供了离散对应。
英文摘要
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.
