Patterns in surface water waves

Walter Craig
McMaster University

Abstract:
I will describe an existence theorem for traveling waves in
water. The first such in two dimensional settings is due to
T. Levi-Civita and D. Struik in the 1920's. In a recent paper
we prove a general result for three dimensions (well, for
any  number of dimensions), when there is surface tension.
The approach is surprisingly close to the Lyapunov center
theorem of A. Weinstein and J. Moser, using the fact due to
V.E. Zakharov that the water waves problem can be formulated
as a Hamiltonian system with infinitely many degrees of
freedom. Without surface tension the problem exhibits small
divisors, and is more difficult.