Mass-Energy threshold dynamics for dipolar Quantum Gases

Abstract

We consider a Gross-Pitaevskii equation which appears as a model in the description of dipolar Bose-Einstein condensates, without a confining external trapping potential. We describe the asymptotic dynamics of solutions to the corresponding Cauchy problem in the energy space in different configurations with respect to the mass-energy threshold, namely for initial data above and at the mass-energy threshold. We first establish a scattering criterion for the equation that we prove by means of the concentration/compactness and rigidity scheme. This criterion enables us to show the energy scattering for solutions with data above the mass-energy threshold, for which only blowup was known. We also prove a blow-up/grow-up criterion for the equation with general data in the energy space. As a byproduct of scattering and blow-up criteria, and the compactness of minimizing sequences for the Gagliardo-Nirenberg's inequality, we study long time dynamics of solutions with data lying exactly at the mass-energy threshold.

Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates

“…As said above, similarly to the classical cubic focusing NLS, if we do not assume any additional hypothesis on the initial datum, as in Theorem 1.2 for example, we cannot prove that the solutions blow-up in finite time. Nonetheless, in [14], Dinh, Hajaiej, and the second author proved the following.…”

“…Nonetheless, we are able to prove a non-linear profile decomposition theorem along bounded non-linear flows, which overcome the lack of finiteness of the scattering norm of the non-linear profiles. See [14,Lemma 3.1]. The latter result in [14] was inspired to [24], where the NLS case was treated.…”

“…The counterpart of Theorem 1.5 is the following scattering result, given for initial data satisfying (1.17). It is one of the main theorems contained in the paper by Dinh, Hajaiej, and the second author [14].…”

“…respectively. Initial data satisfying (1.17) or (1.18) can be constructed by a simple scaling argument, see [14] and [22].…”

“…In [14], Dinh, Hajaiej, and the second author, gave a complete picture of the dynamics under the hypothesis (1.19), by analysing different scenario described in terms of the quantity H(u 0 )M (u 0 ).…”