Wave Breaking in a Class of Nonlocal Dispersive Wave Equations

Abstract

The Korteweg de Vries (KdV) equation is well known as an approximation model for small amplitude and long waves in di!erent physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. In this work we consider a class of nonlocal dispersive wave equations which also incorporate physics of short wavelength scales. The model is identified by a renormalization of an infinite dispersive di!erential operator, followed by further specifications in terms of conservation laws associated with the underlying equation. Several well-known models are thus rediscovered. Wave breaking criteria are obtained for several models including the Burgers-Poisson system, the Camassa-Holm type equation and an Euler-Poisson system. The wave breaking criteria for these models are shown to depend only on the negativity of the initial velocity slope relative to other global quantities.