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We study travelling wave solutions, that is, solutions of the form $v(t, x) =e^{i\lambda t}u(g(t)x)$, to nonlinear Schr\"odinger and Klein-Gordon equationson Riemannian manifolds, both compact and non-compact ones, with emphasis onthe NLKG. Here $g(t)$ represents a one-parameter family of isometries generatedby a Killing field $X$ and a case of particular interest is when $X$ has length$\leq 1$, which leads in certain settings to hypoelliptic operators with lossof at least one derivative. In the compact case, we establish existence oftravelling wave solutions via "energy" minimization methods and prove that atleast compact isotropic manifolds have \emph{genuinely} travelling waves. Weestablish certain sharp regularity estimates on low dimensional spheres thatimprove results in ~\cite{T1} and carry out the subelliptic analysis for NLKGon spheres of higher dimensions utilizing their homogeneous coset spaceproperties. Such subelliptic phenomenon have no parallel in the setting of flatspaces. We will also study related phenomenon on complete noncompact manifoldswith certain symmetry assumptions using concentration-compactness typearguments.

Nonlinear travelling waves on non-Euclidean spaces