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Jon Wilkening, UC BerkeleyComputation of timeperiodic solutions of nonlinear PDE Abstract: I will describe a spectrally accurate numerical method for finding nontrivial timeperiodic solutions of nonlinear PDE. We minimize a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We use adjoint methods (originally developed for shape optimization in fluid mechanics) to compute the gradient of this functional with respect to the initial condition. We then minimize the functional using a quasiNewton gradient descent method. As an application, we study global paths of timeperiodic solutions connecting stationary and traveling waves of the BenjaminOno equation, which is a model water wave equation closely related to the Kortevegde Vries equation. We also study families of timeperiodic symmetric breathing waves for the vortex sheet with surface tension between two incompressible, irrotational, inviscid fluids. This is joint work with David Ambrose.
