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We study the local dynamics near general unstable traveling waves of the 3DGross-Pitaevskii equation in the energy space by constructing smooth localinvariant center-stable, center-unstable and center manifolds. We also provethat (i) the center-unstable manifold attracts nearby orbits exponentiallybefore 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 forwardflow will be ejected away from traveling waves exponentially fast. Furthermore,under a non-degenerate assumption, we show the orbital stability of thetraveling waves on the center manifolds, which also implies the localuniqueness of the local invariant manifolds. Our approach based on a geometricbundle coordinates should work for a general class of Hamiltonian PDEs.

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