Graduate Student / Postdoc Seminar
Steady Water Waves
Speaker: Miles Wheeler
Location: Warren Weaver Hall 1302
Date: Friday, February 27, 2015, 1 p.m.
Having calculated the first five terms in a small-amplitude expansion of periodic traveling water waves, Stokes conjectured in 1880 that such waves could be 'pushed to the extent of yielding crests with sharp edges'. In this mostly expository talk, I will introduce the mathematical theory of steady traveling water waves, in particular the proof of Stokes' conjecture in the eighties using bifurcation theory techniques. Local bifurcation theory allows us to solve equations near points where the implicit function theorem fails, while global bifurcation theory tells us about the possible limiting behavior along branches of solutions; both will be illustrated with simple finite-dimensional examples. I will also review classical results on the symmetry, elevation, and speed of solitary waves, as well as the surprisingly useful reformulation of the steady water wave problem as an ill-posed dynamical system (with the horizontal variable playing the role of time).