On Dipolar Quantum Gases in the Unstable Regime

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Abstract
We study the nonlinear Schrödinger equation arising in dipolar Bose--Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both cases we first focus on the existence and stability/instability properties of standing waves. Our approach leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the free case, by showing that the constrained functional has a so-called mountain pass geometry, we prove the existence of standing states with least energy, the ground states, and show that any ground state is orbitally unstable. Moreover, when the system is free, we show that small data in the energy space scatter in all regimes, stable and unstable. In the second case, if the trapping potential is small, we prove that two different kinds of standing waves appear: one corresponds to a topological local minimizer of the constrained energy functional ...