Abstract
We show that wave breaking occurs with positive probability for the Stochastic Camassa–Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space–time paths.
Highlights
Steepening Lemma: the mechanism for peakon formationIn the following we will continue working on the entire real line R, similar results are available for a periodic domain with only minimal effort
The deterministic CH equation, derived in [2], is a nonlinear shallow water wave equation for a fluid velocity solution whose profile u(x, t) and its gradient both decay to zero at spatial infinity, |x| → ∞, on the real line R
Which emerge from smooth confined initial conditions for the velocity profile. Such a sum is an exact solution of the CH equation (1.1) provided the time-dependent parameters {pa} and {qa}, a = 1, . . . , M, satisfy certain canonical Hamiltonian equations, to be discussed later
Summary
In the following we will continue working on the entire real line R, similar results are available for a periodic domain with only minimal effort. Suppose the initial condition is anti-symmetric, so the inflection point at u = 0 is fixed and dx/dt = 0, due to the symmetry (u, x) → (−u, −x) admitted by equation (1.1) In this case, the total momentum vanishes, i.e. M = 0, and no matter how small |s(0)| (with s(0) < 0), the verticality s → −∞ develops at x in finite time. This process leaves a profile behind with an inflection point of negative slope; so it repeats, thereby producing a wave train of peakons with the tallest and fastest ones moving rightward in order of height. The noise introduced in (2.2) and (2.3) represents an additional stochastic perturbation in the momentum transport velocity
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