Wave breaking for the Whitham equation with fractional dispersion

TL;DR

This paper demonstrates that solutions to the Whitham equation with fractional dispersion can experience wave breaking, where the wave remains bounded but its slope becomes infinite in finite time, under certain initial conditions.

Contribution

It establishes wave breaking phenomena for the Whitham equation with fractional dispersion, expanding understanding of nonlinear wave behavior in this context.

Findings

Wave breaking occurs for sufficiently steep initial data.

Solutions stay bounded while their slopes blow up.

Finite time singularity formation is proven for fractional dispersion cases.

Abstract

We show wave breaking for the Whitham equation in a range of fractional dispersion, i.e. the solution remains bounded but its slope becomes unbounded in finite time, provided that the initial datum is sufficiently steep.