TY - JOUR

T1 - Global bifurcation of solitary waves for the Whitham equation

AU - Truong, Tien

AU - Wahlén, Erik

AU - Wheeler, Miles

N1 - Funding Information:
This research is supported by the Swedish Research Council, grant no. 2016-04999. Part of it was carried out during the workshop Nonlinear water waves—an interdisciplinary interface in 2017 at Erwin Schrödinger International Institute for Mathematics and Physics. The hospitality and support of the institute is gratefully acknowledged. We also thank Grégory Faye and Arnd Scheel for the swift and helpful correspondence during the finishing phase of this project. Finally, we thank the referee for helpful comments.

PY - 2022/8/31

Y1 - 2022/8/31

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85112059051&partnerID=8YFLogxK

U2 - 10.1007/s00208-021-02243-1

DO - 10.1007/s00208-021-02243-1

M3 - Article

SN - 0025-5831

VL - 383

SP - 1521

EP - 1565

JO - Mathematische Annalen

JF - Mathematische Annalen

ER -