Traveling Wave Solutions to NLS and NLKG Equations in Non-Euclidean Settings

We study traveling wave solutions to nonlinear Schrödinger (NLS) and nonlinear Klein-Gordon (NLKG) equations on a compact Riemannian manifold M , with a Killing field X, generating a group of isometries. The emphasis is on NLKG; then if X has length < 1 everywhere, one gets a semilinear elliptic PDE on M , to which standard variational techniques apply (for a natural class of nonlinearities), as reviewed in §1, though there remains the question of whether the associated waves are really (or just apparently) traveling, a point taken up in §2. In §§3–4 we consider sonic speed waves, in some situations that lead to subelliptic nonlinear PDE, and in §5 we consider some supersonic traveling waves.