Mathematics Colloquia and Seminars

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I will describe a spectrally accurate numerical method for finding non-trivial time-periodic 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 quasi-Newton gradient descent method. As an application, we study global paths of time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation, which is a model water wave equation closely related to the Korteveg-de Vries equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). If time permits, I will also discuss our current efforts to find time-periodic solutions of the vortex sheet with surface tension between two incompressible, irrotational, inviscid fluids. This is joint work with David Ambrose.