Global bifurcation of solitary waves for the Whitham equation

TL;DR

This paper proves the existence of a highest, cusped solitary wave solution for the Whitham equation, confirming a long-standing conjecture using advanced bifurcation theory and nonlocal analysis.

Contribution

It extends the bifurcation analysis of the Whitham equation to solitary waves, including the existence of a highest wave with a cusp, using novel nonlocal center manifold techniques.

Findings

Existence of a highest cusped solitary wave confirmed.

Global bifurcation curve approaches the highest wave.

New methods handle singular small-amplitude limit.

Abstract

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnstr\"om and Wahl\'en. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.