916 research outputs found
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
Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type
We prove local and global well-posedness for semi-relativistic, nonlinear
Schr\"odinger equations with
initial data in , . Here is a critical
Hartree nonlinearity that corresponds to Coulomb or Yukawa type
self-interactions. For focusing , which arise in the quantum theory of
boson stars, we derive a sufficient condition for global-in-time existence in
terms of a solitary wave ground state. Our proof of well-posedness does not
rely on Strichartz type estimates, and it enables us to add external potentials
of a general class.Comment: 18 pages; replaced with revised version; remark and reference on blow
up adde
Distributed Nested Rollout Policy for Same Game
Nested Rollout Policy Adaptation (NRPA) is a Monte Carlo search heuristic for puzzles and other optimization problems. It achieves state-of-the-art performance on several games including SameGame. In this paper, we design several parallel and distributed NRPA-based search techniques, and we provide a number of experimental insights about their execution. Finally, we use our best implementation to discover 15 better scores for 20 standard SameGame boards
On the propagation of an optical wave in a photorefractive medium
The aim of this paper is first to review the derivation of a model describing
the propagation of an optical wave in a photorefractive medium and to present
various mathematical results on this model: Cauchy problem, solitary waves
On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations
In our previous paper [12] (Rev. Math. Phys. 16, 383-420 (2004)), a general
framework was outlined to treat the approximate solutions of semilinear
evolution equations; more precisely, a scheme was presented to infer from an
approximate solution the existence (local or global in time) of an exact
solution, and to estimate their distance. In the first half of the present work
the abstract framework of \cite{uno} is extended, so as to be applicable to
evolutionary PDEs whose nonlinearities contain derivatives in the space
variables. In the second half of the paper this extended framework is applied
to theincompressible Navier-Stokes equations, on a torus T^d of any dimension.
In this way a number of results are obtained in the setting of the Sobolev
spaces H^n(T^d), choosing the approximate solutions in a number of different
ways. With the simplest choices we recover local existence of the exact
solution for arbitrary data and external forces, as well as global existence
for small data and forces. With the supplementary assumption of exponential
decay in time for the forces, the same decay law is derived for the exact
solution with small (zero mean) data and forces. The interval of existence for
arbitrary data, the upper bounds on data and forces for global existence, and
all estimates on the exponential decay of the exact solution are derived in a
fully quantitative way (i.e., giving the values of all the necessary constants;
this makes a difference with most of the previous literature). Nextly, the
Galerkin approximate solutions are considered and precise, still quantitative
estimates are derived for their H^n distance from the exact solution; these are
global in time for small data and forces (with exponential time decay of the
above distance, if the forces decay similarly).Comment: LaTeX, 84 pages. The final version published in Reviews in
Mathematical Physic
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
Finite-time blowup for a complex Ginzburg-Landau equation
We prove that negative energy solutions of the complex Ginzburg-Landau
equation blow up in finite time,
where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value , we
obtain estimates of the blow-up time as . It turns out that stays bounded (respectively, goes to
infinity) as in the case where the solution of the
limiting nonlinear Schr\"odinger equation blows up in finite time
(respectively, is global).Comment: 22 page
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
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
Solitary wave dynamics in time-dependent potentials
We rigorously study the long time dynamics of solitary wave solutions of the
nonlinear Schr\"odinger equation in {\it time-dependent} external potentials.
To set the stage, we first establish the well-posedness of the Cauchy problem
for a generalized nonautonomous nonlinear Schr\"odinger equation. We then show
that in the {\it space-adiabatic} regime where the external potential varies
slowly in space compared to the size of the soliton, the dynamics of the center
of the soliton is described by Hamilton's equations, plus terms due to
radiation damping. We finally remark on two physical applications of our
analysis. The first is adiabatic transportation of solitons, and the second is
Mathieu instability of trapped solitons due to time-periodic perturbations.Comment: 38 pages, some typos corrected, one reference added, one remark adde
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