Nonlinear travelling waves on complete Riemannian manifolds (Q1688004)

The paper deals with travelling wave solutions to the nonlinear Schrödinger equation \[ i\partial_tv+\Delta v=-K|v|^{p-1}v, \] and to the nonlinear Klein-Gordon equation \[ \partial^2_tv-\Delta v+m^2v=K|v|^{p-1}v \] over a complete non-compact Riemannian manifold \(M\), which have a bounded Killing field \(X\) that flows by a one-parameter family of isometries \(g(t)\) of \(M\). The author uses standard variational techniques to get existence of travelling waves, \(v(t,x)=e^{i\lambda t}u(g(t)x),\) \(\lambda\in\mathbb{R}\), in case of a complete weakly homogeneous manifold \(M\). If \(M\) is weakly isotropic, the existence of genuine subsonic travelling waves is obtained, at least for a nonempty set of parameters. Moreover, it is proved that a slight perturbation of the Killing field \(X\) leads to a controlled (in suitable \(L^p\)-norms) perturbation of the travelling wave solutions.