Mathematics Colloquium

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Nearly Time-Periodic Water Waves

Speaker: Jon Wilkening, UC Berkeley

Location: Warren Weaver Hall 1302

Date: Monday, March 4, 2013, 3:45 p.m.

Synopsis:

We develop an overdetermined shooting algorithm to compute new families of timeperiodic and quasi-periodic solutions of the free-surface Euler equations involving traveling-standing waves and collisions of solitary waves of various types. The wave amplitudes are too large to be well-approximated by weakly nonlinear theory, yet we observe behavior that resembles elastic collisions of solitons in integrable model equations. A Floquet analysis shows that many of the new solutions are stable to harmonic perturbations. Evolving such perturbations over tens of thousands of cycles suggests that the solutions remain nearly time-periodic forever. We also discuss resonance and re-visit a long-standing conjecture of Penney and Price that the standing water wave of greatest height should form wave crests with sharp, 90 degree interior corner angles.