Nonlinear quasimodes near elliptic periodic geodesics

We consider the nonlinear Schr"odinger equation on a compact manifold near an elliptic periodic geodesic. Using a geometric optics construction, we construct quasimodes to a nonlinear stationary problem which are highly localized near the periodic geodesic. We show the nonlinear Schr"odinger evolution of such a quasimode remains localized near the geodesic, at least for short times.