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

• We consider the Whitham equation u t +2uu x +Lu x =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 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.

Subject headings

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Keyword

Full-dispersion models

Global bifurcation

Highest waves

Whitham equation

Publication and Content Type

art (subject category)

ref (subject category)