Waves of greatest height

Vertr.-Prof. Dr. Gabriele Brüll (Universität Kassel)


In this talk, I will present the phenomenon of so-called waves of greatest height for nonlocal dispersive equations of the form

        ut + Lux + f(u)x = 0,

where L is a Fourier multiplier operator with real symbol m and f is a nonlinearity. In 1880 Stokes conjectured the existence of a traveling surface wave of greatest height, how he called it, for the water-wave problem. He predicted a corner singularity at the crests of the periodic traveling wave, enclosing an angle of precisely 120°. Stokes’ conjecture was affirmed about 100 years later by Amick, Fraenkel, and Toland. Naturally, the question arises, whether water-wave model equations, such as the famous Korteweg–de Vries equation, the Benjamin–Ono equation, the Whitham equation, or the reduced Ostrovsky equation, capture the phenomenon of Stokes’ wave of greatest height. Many of the water-wave model equations take the form above, including the four mentioned ones. I will present some results of joint works with R. Dhara (Brno), M. Ehrnström (Trondheim), and L. Pei (Zhuhai), as well as open problems.

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