Nonlinear waves in open and closed systems: dispersion vs. weak turbulence

Dynamics of weakly nonlinear waves (such as small amplitude water or gravitational waves) depends crucially on whether the domains in which the waves propagate are unbounded (open) or bounded (closed). On unbounded domains the evolution of waves is stabilized by the dissipation of energy by dispersion, while on bounded domains the evolution is destabilized by weak turbulence. In my talk, I will illustrate these two kinds of behaviour first with simple examples and then with solutions of Einstein's equations.