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Abstract
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The Gross-Pitaevskii equations are widely used in modeling superfluids and Bose-Einstein condensates. The GP equation has traveling waves solutions which have non-vanishing limit at infinity, first discovered by physicists (Jones, Roberts et al., 1980s). The existence of such traveling waves has been studied a lot in recent years by Betheul, Saut and many others. However, the stability and dynamical behaviors near such traveling waves are not well understood. With Zhengping Wang and Chongchun Zeng, we proved a nonlinear stability criterion for 3D traveling waves as conjectured in the physical literature, under a non-degeneracy assumption. Moreover, the invariant manifolds (stable, unstable and center) are constructed near unstable traveling waves, and the orbital stability is proved on center manifold.
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