Department of

Mathematics


Seminar Calendar
for events the day of Wednesday, December 15, 2004.

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Wednesday, December 15, 2004

12:00 pm in 245 Altgeld Hall,Wednesday, December 15, 2004

Resonance Phenomena in Nonlinear Dispersive Partial Differential Equations

Eduard-Wilhelm Kirr (University of Chicago)

Abstract: Abstract: While the resonance phenomena and stability of a periodically forced, linear oscillator is well understood, the problem becomes quite difficult when the mechanical system has more than one degree of freedom and the forcing depends on the state of the system. Multiple scale analysis, Poincare continuation and KAM theory give only partial answers. My talk will focus on recent, rigorous results concerning systems with infinitely many degrees of freedom. I will briefly describe why such systems are ubiquitous in Quantum Mechanics, Statistical Physics and Optics where they are modeled by dispersive partial differential equations. A simplified mechanical example would be a mass-spring system suspended on an infinitely long, tense string. The oscillations of the spring excites (resonantly) the string which carries the energy of the excitations to infinity. As a result one sees a decay of the amplitude with which the mass-spring system oscillates. I will present in some detail the mathematical techniques involved in proving that the same phenomenon occurs for the ground state of the cubic nonlinear Schroedinger equation subject to periodic in time perturbation, a result obtained in collaboration with S. Cuccagna and D. Pelinovsky. Then I will connect this result with the ones for random and almost periodic perturbations of linear Hamiltonian partial differential equations obtained in collaboration with M. Weinstein. At the end I will mention some related open problems and argue that the above results and mathematical techniques developed constitute a solid basis for attacking them.