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

This paper investigates the stochastic Camassa–Holm equation with additive noise as a model for nonlinear shallow-water waves under random external forcing. It establishes that finite-time breakdown almost surely occurs through wave breaking, with bounded wave amplitude and positive slope but a minimum slope tending to negative infinity. Under an additional moment assumption, the authors derive an initial-slope condition.