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2011 Wolfe Lecture in Mathematics

Dr. Wilhelm Schlag
University of Chicago


Thursday April 7, 2011
4:00pm, Herman Brown 227


Title: Center manifolds and dispersive Hamiltonian evolution equations

 

Abstract: By means of certain dispersive PDEs (such as the nonlinear Klein-Gordon equation) we will exhibit a new family of phenomena related to the ground state solitons. These solitons are exponentially unstable, and one can construct stable, unstable, and center(-stable) manifolds associated with these ground states in the sense of hyperbolic dynamics. In terms of these manifolds one can completely characterize the global dynamics of solutions whose energy exceeds that of the ground states by at most a small amount. In particular, we will establish a trichotomy in forward time giving either finite-time blow up, global forward existence and scattering to zero, or global existence and scattering to the ground states as all possibilities. It turns out that all nine sets consisting of all possible combinations of the forward/backward trichotomies arise.

This extends the classical Payne-Sattinger picture (from 1975) which gives such a characterization at energies below that of the ground state; in the latter case the aforementioned (un)stable and center manifolds do not arise, since they require larger energy than that of the ground state. Our methods proceed by combining a perturbative analysis near the ground states with a global and variational analysis away from them.

Most of this work is joint with Kenji Nakanishi from Kyoto University, Japan.

 

 

The Annual Wolfe Lectures are supported through a generous gift from the estate of Raquel A. Wolfe in loving memory of her husband, Dr. Alfred S. Wolfe, and his love for math which he instilled in both his children.

 

Updated July 1, 2014