Invariant Manifolds of Traveling Waves of the 3D Gross–Pitaevskii Equation in the Energy Space

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 go away from the traveling waves along the center or unstable directions and (ii) if an initial data is not on the center-stable manifolds, then the forward orbit leaves traveling waves exponentially fast. Furthermore, under an additional non-degeneracy assumption, we show the orbital stability of the travelingwaves on the centermanifolds,which also implies the uniqueness of the local invariant manifolds. Our method based on a geometric bundle coordinate system should work for a general class of Hamiltonian PDEs.