Nonlinear travelling waves on non-Euclidean spaces

Abstract

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 equations on Riemannian manifolds, both compact and non-compact ones, with emphasis on the NLKG. Here $g(t)$ represents a one-parameter family of isometries generated by 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 loss of at least one derivative. In the compact case, we establish existence of travelling wave solutions via "energy" minimization methods and prove that at least compact isotropic manifolds have \emph{genuinely} travelling waves. We establish certain sharp regularity estimates on low dimensional spheres that improve results in ~\cite{T1} and carry out the subelliptic analysis for NLKG on spheres of higher dimensions utilizing their homogeneous coset space properties. Such subelliptic phenomenon have no parallel in the setting of flat spaces. We will also study related phenomenon on complete noncompact manifolds with certain symmetry assumptions using concentration-compactness type arguments.