Invariant Manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space

TL;DR

This paper analyzes the local dynamics near unstable traveling waves of the 3D Gross-Pitaevskii equation, constructing invariant manifolds and proving stability and attraction properties in the energy space.

Contribution

It introduces a geometric bundle coordinate approach to construct invariant manifolds and establishes stability results for traveling waves in a general Hamiltonian PDE setting.

Findings

Center-unstable manifold attracts nearby orbits exponentially.

Initial data off the center-stable manifold is ejected exponentially fast.

Traveling waves are orbitally stable on the center manifolds under a non-degenerate assumption.

Abstract

We study the local dynamics near general unstable traveling waves of the 3D Gross-Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the center-unstable manifold attracts nearby orbits exponentially before they get away from the traveling waves along the center directions and (ii) if an initial data is not on the center-stable manifolds, then the forward flow will be ejected away from traveling waves exponentially fast. Furthermore, under a non-degenerate assumption, we show the orbital stability of the traveling waves on the center manifolds, which also implies the local uniqueness of the local invariant manifolds. Our approach based on a geometric bundle coordinates should work for a general class of Hamiltonian PDEs.