Nonlinear travelling waves on complete Riemannian manifolds

Abstract

We study travelling wave solutions to nonlinear Schrödinger and Klein-Gordon equations on complete Riemannian manifolds, which have a bounded Killing field $X$. For a natural class of power-type nonlinearities, we use standard variational techniques to demonstrate the existence of travelling waves on complete weakly homogeneous manifolds. If the manifolds in question are weakly isotropic, we prove that they have genuine subsonic travelling waves, at least for a non-empty set of parameters. Finally we establish that a slight perturbation of the Killing field $X$ will result in a controlled perturbation of the travelling wave solutions (in appropriate $L^p$-norms).