Periodic and Quasiperiodic Waves on the Sphere

We construct periodic and quasiperiodic solutions for the nonlinear plate equation on the unit sphere in \BbbR3. Periodic solutions for the nonlinear wave equation are constructed as well. The methods do not involve small parameters. In the case of a cubic nonlinear term, our approach is computer-assisted and thus yields detailed information about each solution. For nonlinearities that are in some sense close to linear, weak solutions are obtained by a variational method.