Wave breaking in the Whitham equation

TL;DR

This paper proves wave breaking phenomena in a nonlocal water wave model, confirming a longstanding conjecture and extending results to fractional dispersion equations, highlighting conditions for finite-time singularity formation.

Contribution

It establishes the occurrence of wave breaking in the Whitham equation and related models, solving Whitham's conjecture and broadening understanding of nonlocal wave dynamics.

Findings

Wave breaking occurs for sufficiently negative initial slopes.

The result confirms Whitham's conjecture on wave breaking.

Extension to fractional KdV-type equations is achieved.

Abstract

We prove wave breaking --- bounded solutions with unbounded derivatives --- in the nonlinear nonlocal equation which combines the dispersion relation of water waves and a nonlinearity of the shallow water equations, provided that the slope of the initial datum is sufficiently negative, whereby we solve a Whitham's conjecture. We extend the result to equations of Korteweg-de Vries type for a range of fractional dispersion.