Nonlinear quasimodes near elliptic periodic geodesics

TL;DR

This paper constructs highly localized nonlinear quasimodes near elliptic periodic geodesics on compact manifolds and demonstrates their stability under nonlinear Schrödinger evolution for short times.

Contribution

It introduces a geometric optics method to build nonlinear quasimodes localized near elliptic geodesics and analyzes their evolution.

Findings

Construction of nonlinear quasimodes localized near geodesics

Demonstration of their stability under Schrödinger evolution

Short-time persistence of localization

Abstract

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.