102 research outputs found
Existence of solitary waves in dipolar quantum gases
We study a nonlinear Schroedinger equation arising in the mean-field
description of dipolar quantum gases. Under the assumption of sufficiently
strong dipolar interactions, the existence of standing waves, and hence
solitons, is proved together with some of their properties. This gives a
rigorous argument for the possible existence of solitary waves in Bose-Einstein
condensates, which originate solely due to the dipolar interaction between the
particles.Comment: Minor modifications; more explanations added. To appear in Physica
Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement
We consider the three-dimensional time-dependent Gross-Pitaevskii equation
arising in the description of rotating Bose-Einstein condensates and study the
corresponding scaling limit of strongly anisotropic confinement potentials. The
resulting effective equations in one or two spatial dimensions, respectively,
are rigorously obtained as special cases of an averaged three dimensional limit
model. In the particular case where the rotation axis is not parallel to the
strongly confining direction the resulting limiting model(s) include a
negative, and thus, purely repulsive quadratic potential, which is not present
in the original equation and which can be seen as an effective centrifugal
force counteracting the confinement.Comment: 22 page
Global attractor for a Ginzburg-Landau type model of rotating Bose-Einstein condensates
We study the long time behavior of solutions to a nonlinear partial
differential equation arising in the description of trapped rotating
Bose-Einstein condensates. The equation can be seen as a hybrid between the
well-known nonlinear Schr\"odinger/Gross-Pitaevskii equation and the
Ginzburg-Landau equation. We prove existence and uniqueness of global in-time
solutions in the physical energy space and establish the existence of a global
attractor within the associated dynamics. We also obtain basic structural
properties of the attractor and an estimate on its Hausdorff and fractal
dimensions.Comment: 25 pages; some more typos fixed; additional references adde
Geometric optics and instability for NLS and Davey-Stewartson models
We study the interaction of (slowly modulated) high frequency waves for
multi-dimensional nonlinear Schrodinger equations with gauge invariant
power-law nonlinearities and non-local perturbations. The model includes the
Davey--Stewartson system in its elliptic-elliptic and hyperbolic-elliptic
variant. Our analysis reveals a new localization phenomenon for non-local
perturbations in the high frequency regime and allows us to infer strong
instability results on the Cauchy problem in negative order Sobolev spaces,
where we prove norm inflation with infinite loss of regularity by a
constructive approach.Comment: 33 page
Stability and instability properties of rotating Bose-Einstein condensates
We consider the mean-field dynamics of Bose-Einstein condensates in rotating
harmonic traps and establish several stability and instability properties for
the corresponding solution. We particularly emphasize the difference between
the situation in which the trap is symmetric with respect to the rotation axis
and the one where this is not the case.Comment: 13 pages; several typos corrected; new references added; more details
given in some of the proof
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