Numerical Solution of the Small Dispersion Limit of the Camassa–Holm and Whitham Equations and Multiscale Expansions

The small dispersion limit of solutions to the Camassa–Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture. We present a quantitative numerical comparison between the solution to the CH equation and the asymptotic solution. The dependence on the small dispersion parameter $epsilon$ is studied in the interior and at the boundaries of the Whitham zone. In the interior of the zone, the difference between the solution to the CH equation and the asymptotic solution is of the order $epsilon$, at the trailing edge of the order $sqrt{epsilon}$, and at the leading edge of the order $epsilon^{1/3}$. For the latter we present a multiscale expansion which describes the amplitude of the oscillations in terms of the Hastings–McLeod solution of the Painlevé II equation. We show numerically that this multiscale solution provides an enhanced asymptotic description near the leading edge.