Abstract

The long-time behavior of nonlinear dispersive waves subject to spatial confinement can be very rich and complex because, in contrast to unbounded domains, waves cannot escape to infinity and keep self-interacting for all times. If, in addition, the linear spectrum around the ground state is fully resonant, then the nonlinearity can produce significant effects for arbitrarily small perturbations. The weak field dynamics of such systems can be approximated by solutions of the corresponding infinite-dimensional time-averaged Hamiltonian systems which govern resonant interactions between the modes. A major mathematical challenge in this context is to describe the energy transfer between the modes. I will discuss this problem for several spatially confined systems: a cubic wave equation on the 3-sphere, the Einstein-scalar system with negative cosmological constant (modeling the weakly turbulent behavior of small perturbations of the anti-de Sitter spacetime), the nonlinear Schroedinger equation with a trapping potential (modeling the dynamics of a Bose-Einstein condensate), and the Schroedinger-Newton-Hooke system describing a harmonically trapped self-gravitating condensate. Some intriguing parallels between these systems will be emphasized.