On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation

Ehrnström, Mats; Wahlén, Erik

doi:10.1016/j.anihpc.2019.02.006

¹Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
²Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1603-1637

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Résumé

We consider the Whitham equation $u_{t} + 2 u u_{x} + L u_{x} = 0$ , where L is the nonlocal Fourier multiplier operator given by the symbol $m (ξ) = \sqrt{\tanh ξ / ξ}$ . G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of P-periodic solutions, and give several qualitative properties of it, including its optimal $C^{1 / 2}$ -regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m (ξ)$ , and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol $m (ξ)$ is completely monotone, and provide an explicit representation formula for it. Our methods may be generalised.

MR Zbl

DOI : 10.1016/j.anihpc.2019.02.006

Keywords: Whitham equation, Full-dispersion models, Highest waves, Global bifurcation

Affiliations des auteurs :

Ehrnström, Mats ¹ ; Wahlén, Erik ²

¹ Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
² Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden

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     title = {On {Whitham's} conjecture of a highest cusped wave for a nonlocal dispersive equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1603--1637},
     year = {2019},
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Ehrnström, Mats; Wahlén, Erik. On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1603-1637. doi: 10.1016/j.anihpc.2019.02.006

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