Dynamical collapse of cylindrical symmetric Dipolar Bose-Einstein condensates

Abstract

We study the formation of singularities for cylindrical symmetric solutions to the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate. We prove that solutions arising from initial data with energy below the energy of the Ground State and that do not scatter collapse in finite time. The main tools to prove our result are the variational characterization of the Ground State energy, suitable localized virial identities for cylindrical symmetric functions, and general integral and pointwise estimates for operators involving powers of the Riesz transforms.

34 pages, final version. Calculus of Variations and Partial Differential Equations, to appear

References in corpus (5)

• 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
• On the Gross-Pitaevskii equation for trapped dipolar quantum gases
• Asymptotic dynamic for dipolar Quantum Gases below the ground state energy threshold
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Cited by in corpus (5)

• Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with ${\chi }^3$ nonlinear response
• On finite time blow-up for a 3D Davey-Stewartson system
• Blowup of cylindrically symmetric solutions for biharmonic NLS
• Scattering of the energy-critical NLS with dipolar interaction
• Standing waves for a Schrödinger system with three waves interaction