28 research outputs found
Global well-posedness for a nonlocal Gross-Pitaevskii equation with non-zero condition at infinity
We study the Gross-Pitaevskii equation involving a nonlocal interaction
potential. Our aim is to give sufficient conditions that cover a variety of
nonlocal interactions such that the associated Cauchy problem is globally
well-posed with non-zero boundary condition at infinity, in any dimension. We
focus on even potentials that are positive definite or positive tempered
distributions.Comment: Communications in Partial Differential Equations (2010
Traveling waves for nonlinear Schr\"odinger equations with nonzero conditions at infinity, II
We prove the existence of nontrivial finite energy traveling waves for a
large class of nonlinear Schr\"odinger equations with nonzero conditions at
infinity (includindg the Gross-Pitaevskii and the so-called "cubic-quintic"
equations) in space dimension . We show that minimization of the
energy at fixed momentum can be used whenever the associated nonlinear
potential is nonnegative and it gives a set of orbitally stable traveling
waves, while minimization of the action at constant kinetic energy can be used
in all cases. We also explore the relationship between the families of
traveling waves obtained by different methods and we prove a sharp nonexistence
result for traveling waves with small energy.Comment: Final version, accepted for publication in the {\it Archive for
Rational Mechanics and Analysis.} The final publication is available at
Springer via http://dx.doi.org/10.1007/s00205-017-1131-
Travelling waves for the Gross-Pitaevskii equation II
The purpose of this paper is to provide a rigorous mathematical proof of the
existence of travelling wave solutions to the Gross-Pitaevskii equation in
dimensions two and three. Our arguments, based on minimization under
constraints, yield a full branch of solutions, and extend earlier results,
where only a part of the branch was built. In dimension three, we also show
that there are no travelling wave solutions of small energy.Comment: Final version accepted for publication in Communications in
Mathematical Physics with a few minor corrections and added remark
Minimal energy for the traveling waves of the Landau-Lifshitz equation
We consider nontrivial finite energy traveling waves for the Landau-Lifshitz
equation with easy-plane anisotropy. Our main result is the existence of a
minimal energy for these traveling waves, in dimensions two, three and four.
The proof relies on a priori estimates related with the theory of harmonic maps
and the connection of the Landau-Lifshitz equation with the kernels appearing
in the Gross-Pitaevskii equation.Comment: submitte
The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate
We study the Gross-Pitaevskii (GP) energy functional for a fast rotating
Bose-Einstein condensate on the unit disc in two dimensions. Writing the
coupling parameter as 1 / \eps^2 we consider the asymptotic regime \eps
\to 0 with the angular velocity proportional to
(\eps^2|\log\eps|)^{-1} . We prove that if \Omega = \Omega_0
(\eps^2|\log\eps|)^{-1} and then a minimizer of
the GP energy functional has no zeros in an annulus at the boundary of the disc
that contains the bulk of the mass. The vorticity resides in a complementary
`hole' around the center where the density is vanishingly small. Moreover, we
prove a lower bound to the ground state energy that matches, up to small
errors, the upper bound obtained from an optimal giant vortex trial function,
and also that the winding number of a GP minimizer around the disc is in accord
with the phase of this trial function.Comment: 52 pages, PDFLaTex. Minor corrections, sign convention modified. To
be published in Commun. Math. Phy
Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with non-zero conditions at infinity
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