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. 
Pierre Albin and Hans Christianson and Jeremy L. Marzuola and Laurent   Thomann
math.AP