Time-periodic solutions of the Benjamin–Ono equation

Jon Wilkening, UC Berkeley
September 12th, 2008 at 11AM–12PM in 939 Evans Hall [Map]

I will describe a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear PDE. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin–Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0.

We use our method to study global paths of solutions connecting stationary and traveling waves of the Benjamin–Ono 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). By experimentation with data fitting, we identify the analytic form of these solutions in terms of the trajectories of the Fourier modes. We then prove that solutions of the form suggested by our numerical experiments exist and show how our representation fits into the hierarchy of previously known multi-periodic and multi-soliton solutions.

This is joint work with David Ambrose.

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