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A dipolar Gross-Pitaevskii equation with quantum fluctuations: Self-bound states

Yongming Luo, Athanasios Stylianou

Published Jun 2, 2018• 1 citation
OpenAlex

Abstract

We prove existence and qualitative properties of standing wave solutions to a generalized nonlocal 3rd-4th order Gross-Pitaevskii equation (GPE), the latter being currently the state-of-the-art model for describing the dynamics of dipolar Bose-Einstein condensates. Using a mountain pass argument on spheres in $L^2$ and constructing appropriately localized Palais-Smale sequences we are able to prove existence of real positive ground states as saddle points of the energy. The analysis is deployed in the set of possible states, thus overcoming the problem that the energy is unbounded below. We also prove a corresponding nonlocal Pohozaev identity with no rest term, a crucial part of the analysis.

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Journal of Physics B Atomic Molecular and Optical Physics•Jul 26, 2018•19 citations
We prove existence and qualitative properties of standing wave solutions to a
generalized nonlocal 3rd-4th order Gross-Pitaevskii equation (GPE), the latter
being currently the state-of-the-art model for describing the dynamics of
dipolar Bose-Einstein condensates. Using a mountain pass argument on spheres in
L2L^2L2 and constructing appropriately localized Palais-Smale sequences we are
able to prove existence of real positive ground states as saddle points of the
energy. The analysis is deployed in the set of possible states, thus overcoming
the problem that the energy is unbounded below. We also prove a corresponding
nonlocal Pohozaev identity with no rest term, a crucial part of the analysis.
Abstract. We study the existence and stability / instability properties of standing waves for a 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 add. In both cases this leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the first case we prove the existence and instability of ground states by showing that the constrained functional has a so-called mountain pass geometry. In the second case we prove that, if the trapping potential is small, two different kind of standing waves appears: one corresponds to a local minima of the constrained energy functional, the other is again of mountain pass type. Despite the problem is mass supercritical and the functional unbounded from below the standing waves associated to the local minima are orbitally stable. Eventually we show that the addition of the trapping potential, however small, create a “gap ” in the ground state energy level of the system. 1.
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 and consists in ground states, and the other is again of mountain pass type but now corresponds to excited states. We also prove that any ground state is a topological local minimizer. Despite the problem being mass supercritical and the functional being unbounded from below, the standing waves associated to the set of ground states turn out to be orbitally stable. Actually, from the physical point of view, the introduction of the trapping potential stabilizes the system that is initially unstable. Related to this we observe that it also creates a gap in the ground state energy level of the system. In addition when the trapping potential is active the presence of standing waves with arbitrary small norm does not permit small data scattering. Eventually some asymptotic results are also given.
The nonlinear dynamics of self-bound Bose–Einstein matter waves under the action of an attractive nonlocal and a repulsive local interaction is analyzed by means of a time-dependent variational formalism. The mean-field model described by the Gross–Pitaevskii equation (GPE) is reduced to a single second-order conservative ordinary differential equation admitting a potential function. The stable fixed points and the linear oscillation frequencies of the breather modes are determined. The chosen nonlocal interactions are a Gaussian-shaped potential and a van der Waals-like potential. The variational solutions are compared with direct numerical simulations of the GPE, for each of these nonlocalities. The spatio-temporal frequency spectra of linear waves are also determined.