| Abstract: |
| Stokes in his classical memoir made formal but far-reaching considerations about periodic traveling waves at the surface of water, subject to the force of gravity, and conjectured that the wave of greatest height exhibits a $120$ degree`s corner at the crest. For zero vorticity, Amick, Fraenkel and Toland proved that such a limiting wave exists. But, for nonzero vorticity, the situation is much more complicated.
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I will present a recent work, with Sergey Dyachenko (Illinois), where we formulate the Stokes wave problem, permitting constant vorticity and finite depth, via a conformal mapping as a nonlinear pseudo-differential equation, involving a periodic Hilbert transform for a strip, and solve by the Newton-GMRES method and a fast Fourier transform. We find overhanging and touching waves for strong positive vorticity, and Crapper`s limiting wave and the rigid body rotation of a fluid disk at the large vorticity limit. |
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