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arXiv · 1711.07619

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

Jiayin Jin·Zhiwu Lin·Chongchun Zeng
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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.

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Jiayin Jin, Zhiwu Lin, Chongchun Zeng. 2017-11-21. Invariant Manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space. https://doi.org/10.1007/s00220-018-3189-6

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