Mathematics Colloquia and Seminars

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Description

The resonance phenomena and stability of a periodically forced,linear oscillator is well understood. But the problem becomes quite difficultwhen 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 attached to an infinitely long, tense string. The oscillations of the spring excite (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 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 discuss the similarities and differences between this result and 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 the mathematical techniques developed constitute a solid basis for attacking them.