Asymptotic dynamic for dipolar Quantum Gases below the ground state energy threshold

Abstract

We consider the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate without external confinement. We first consider the unstable regime, where the nonlocal nonlinearity is neither positive nor radially symmetric and standing states are known to exist. We prove that under the energy threshold given by the ground state, all global in time solutions behave as free waves asymptotically in time. The ingredients of the proof are variational characterization of the ground states energy, a suitable profile decomposition theorem and localized virial estimates, enabling to carry out a Concentration/Compactness and Rigidity scheme. As a byproduct we show that in the stable regime, where standing states do not exist, any initial data in the energy space scatters.

arXiv commentFixed typos and minor changes. Journal of Functional Analysis, in press

References in corpus (8)

• Bose-Einstein condensation in trapped dipolar gases
• Mathematical theory and numerical methods for Bose-Einstein condensation
• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates
• On dipolar quantum gases in the unstable regime
• On the Gross-Pitaevskii equation for trapped dipolar quantum gases
• Scattering of two-dimensional solitons in dipolar Bose-Einstein condensates

Cited by in corpus (5)

• Global dynamics below the ground states for NLS under partial harmonic confinement
• Dynamical collapse of cylindrical symmetric Dipolar Bose-Einstein condensates
• On finite time blow-up for a 3D Davey-Stewartson system
• Scattering of the energy-critical NLS with dipolar interaction
• On the stability of 2D dipolar Bose-Einstein condensates