Global bifurcation of travelling gravity waves with vorticity

wwaveOn Thursday afternoon, Eugen Varvaruca spoke on global bifurcation of gravity waves with vorticity.  He first reviewed the various formulations suitable for global bifurcation theories, and settled on Bobenko’s formulation as a starting point.  He introduced a new integro-differential formulation which generalizes Bobenko’s formulation to finite depth and non-zero vorticity.  This formulation can be characterized as the Euler-Lagrange equation for a suitable functional.  A generalization of the periodic Hilbert transform was introduced, and used to define a Dirichlet-Neumann operator of the form w –> [w]/d + C(w’) where C(.) is the generalized Hilbert transform and [.] denotes average.  Real analytic bifurcation theory was used to find branching travelling waves.  First local bifurcations were established in the massflux-energy (Bernoulli) plane.  Then global results were established using Rabinowitz’s theorem.  Several possibilities for global branches were discussed: reconnect to a trivial solution, stagnation points, singularities, but the theory has yet to discriminate which case occurs.  A preprint Constantin, Strauss & Varvaruca (2014) has been posted on the arXiv. A video of the talk is available here.

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