2,734 research outputs found
On a fourth order nonlinear Helmholtz equation
In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz
equation in for positive, bounded and -periodic functions . Using
the dual method of Evequoz and Weth, we find solutions to this equation and
establish some of their qualitative properties
On the cubic NLS on 3D compact domains
We prove bilinear estimates for the Schr\"odinger equation on 3D domains,
with Dirichlet boundary conditions. On non-trapping domains, they match the
case, while on bounded domains they match the generic boundary
less manifold case. As an application, we obtain global well-posedness for the
defocusing cubic NLS for data in , , with
any bounded domain with smooth boundary.Comment: 15 pages, updated references and corrected typos. To appear in
Journal of the Institute of Mathematics of Jussie
On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space
In this paper, we study nonlinear Helmholtz equations (NLH)
in , where
denotes the Laplace-Beltrami operator in the hyperbolic space
and is chosen suitably. Using fixed point
and variational techniques, we find nontrivial solutions to (NLH) for all
and . The oscillatory behaviour and decay rates of radial
solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds
and Damek-Ricci spaces. Our results rely on a new Limiting Absorption Principle
for the Helmholtz operator in . As a byproduct, we obtain simple
counterexamples to certain Strichartz estimates
Growth of Sobolev norms for the quintic NLS on
We study the quintic Non Linear Schr\"odinger equation on a two dimensional
torus and exhibit orbits whose Sobolev norms grow with time. The main point is
to reduce to a sufficiently simple toy model, similar in many ways to the one
used in the case of the cubic NLS. This requires an accurate combinatorial
analysis.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with
arXiv:0808.1742 by other author
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
Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control.
Beauchard and Laurent proved that, under an appropriate non degeneracy
assumption, this system is controllable, locally around the ground state, in
arbitrary time. Coron proved that a positive minimal time is required for this
controllability, on a particular degenerate example.
In this article, we propose a general context for the local controllability
to hold in large time, but not in small time. The existence of a positive
minimal time is closely related to the behaviour of the second order term, in
the power series expansion of the solution
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