Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces

Abstract. We prove the existence of Cantor families of small amplitude periodic solutions for wave and Schrödinger equations on compact Lie groups and homogeneous spaces with merely differentiable nonlin-earities. The NLS equation on homogeneous spaces arises as a mean field approximation of condensates of many-body lattice problems. The highly degenerate eigenvalues of the Laplace Beltrami operator give Nash-Moser implicit function theorem. We provide a new algebraic framework to prove the key tame esti-mates for the inverse linearized operators along Banach scales of Sobolev functions. We exploit properties of the eigenvalues and eigenfunctions of the Laplace Beltrami operator on Lie groups.