Global bifurcation of solitary waves for the Whitham equation

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 Ehrnström and Wahlén. 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.

arXiv commentJournal version. Mathematische Annalen, Online First. 45 pages, 3 figures

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• Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart
• Periodic Hölder waves in a class of negative-order dispersive equations
• Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime
• Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols