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

Ehrnström, Mats; Wahlén, Erik

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

We consider the Whitham equation ut+2uux+Lux=0, where L is the nonlocal Fourier multiplier operator given by the symbol m(ξ)=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 C1/2-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.

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