Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces (Q640820)

In this paper, the authors prove the existence of Cantor families of small amplitude time-periodic solutions for the nonlinear wave equation \(u_{tt} -\Delta u+\mu u=f(x,u)\) and the nonlinear Schrödinger equation \(iu_{t} -\Delta u+\mu u=f(x,|u|^2 )u\), where the spatial variable \(x\) ranges over an arbitrary compact Lie group \(M\) or, more generally, over an arbitrary compact homogeneous space \(M\) (for instance, the sphere \(S^n\)). The operator \(\Delta\) is the Laplace-Beltrami operator on \(M\) with respect to a Riemannian metric compatible with the group structure or action, the mass \(\mu\) is positive, and the nonlinearity is finitely differentiable and vanishes at \(u=0\) at least quadratically. The proof of the existence of these Cantor families of solutions is based on an abstract Nash-Moser implicit function theorem developed by them.