594 research outputs found
Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrodinger system
In this paper, we establish two new types of invariant sets for the coupled
nonlinear Schrodinger system on , and derive two sharp thresholds
of blow-up and global existence for its solution. Some analogous results for
the nonlinear Schrodinger system posed on the hyperbolic space
and on the standard 2-sphere are also presented. Our arguments
and constructions are improvements of some previous works on this direction. At
the end, we give some heuristic analysis about the strong instability of the
solitary waves.Comment: 21 page
On the high-low method for NLS on the hyperbolic space
In this paper, we first prove that the cubic, defocusing nonlinear
Schr\"odinger equation on the two dimensional hyperbolic space with radial
initial data in is globally well-posed and scatters when . Then we extend the result to nonlineraities of order . The
result is proved by extending the high-low method of Bourgain in the hyperbolic
setting and by using a Morawetz type estimate proved by the first author and
Ionescu.Comment: The result is extended to general nonlineraitie
On the density-potential mapping in time-dependent density functional theory
The key questions of uniqueness and existence in time-dependent density
functional theory are usually formulated only for potentials and densities that
are analytic in time. Simple examples, standard in quantum mechanics, lead
however to non-analyticities. We reformulate these questions in terms of a
non-linear Schr\"odinger equation with a potential that depends non-locally on
the wavefunction.Comment: 8 pages, 2 figure
On convergence towards a self-similar solution for a nonlinear wave equation - a case study
We consider the problem of asymptotic stability of a self-similar attractor
for a simple semilinear radial wave equation which arises in the study of the
Yang-Mills equations in 5+1 dimensions. Our analysis consists of two steps. In
the first step we determine the spectrum of linearized perturbations about the
attractor using a method of continued fractions. In the second step we
demonstrate numerically that the resulting eigensystem provides an accurate
description of the dynamics of convergence towards the attractor.Comment: 9 pages, 5 figure
On approximate solutions of semilinear evolution equations
A general framework is presented to discuss the approximate solutions of an
evolution equation in a Banach space, with a linear part generating a semigroup
and a sufficiently smooth nonlinear part. A theorem is presented, allowing to
infer from an approximate solution the existence of an exact solution.
According to this theorem, the interval of existence of the exact solution and
the distance of the latter from the approximate solution can be evaluated
solving a one-dimensional "control" integral equation, where the unknown gives
a bound on the previous distance as a function of time. For example, the
control equation can be applied to the approximation methods based on the
reduction of the evolution equation to finite-dimensional manifolds: among
them, the Galerkin method is discussed in detail. To illustrate this framework,
the nonlinear heat equation is considered. In this case the control equation is
used to evaluate the error of the Galerkin approximation; depending on the
initial datum, this approach either grants global existence of the solution or
gives fairly accurate bounds on the blow up time.Comment: 33 pages, 10 figures. To appear in Rev. Math. Phys. (Shortened
version; the proof of Prop. 3.4. has been simplified
Higher order Schrodinger and Hartree-Fock equations
The domain of validity of the higher-order Schrodinger equations is analyzed
for harmonic-oscillator and Coulomb potentials as typical examples. Then the
Cauchy theory for higher-order Hartree-Fock equations with bounded and Coulomb
potentials is developed. Finally, the existence of associated ground states for
the odd-order equations is proved. This renders these quantum equations
relevant for physics.Comment: 19 pages, to appear in J. Math. Phy
Nonlinear Quantum Mechanics and Locality
It is shown that, in order to avoid unacceptable nonlocal effects, the free
parameters of the general Doebner-Goldin equation have to be chosen such that
this nonlinear Schr\"odinger equation becomes Galilean covariant.Comment: 10 pages, no figures, also available on
http://www.pt.tu-clausthal.de/preprints/asi-tpa/012-97.htm
Collapse of an Instanton
We construct a two parameter family of collapsing solutions to the 4+1
Yang-Mills equations and derive the dynamical law of the collapse. Our
arguments indicate that this family of solutions is stable. The latter fact is
also supported by numerical simulations.Comment: 17 pages, 1 figur
An effective mass theorem for the bidimensional electron gas in a strong magnetic field
We study the limiting behavior of a singularly perturbed
Schr\"odinger-Poisson system describing a 3-dimensional electron gas strongly
confined in the vicinity of a plane and subject to a strong uniform
magnetic field in the plane of the gas. The coupled effects of the confinement
and of the magnetic field induce fast oscillations in time that need to be
averaged out. We obtain at the limit a system of 2-dimensional Schr\"odinger
equations in the plane , coupled through an effective selfconsistent
electrical potential. In the direction perpendicular to the magnetic field, the
electron mass is modified by the field, as the result of an averaging of the
cyclotron motion. The main tools of the analysis are the adaptation of the
second order long-time averaging theory of ODEs to our PDEs context, and the
use of a Sobolev scale adapted to the confinement operator
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