439 research outputs found
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
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
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
A new phase in the production of quality-controlled sea level data
Sea level is an essential climate variable (ECV) that has a direct effect on many people through inundations of coastal areas, and it is also a clear indicator of climate changes due to external forcing factors and internal climate variability. Regional patterns of sea level change inform us on ocean circulation variations in response to natural climate modes such as El Niño and the Pacific Decadal Oscillation, and anthropogenic forcing. Comparing numerical climate models to a consistent set of observations enables us to assess the performance of these models and help us to understand and predict these phenomena, and thereby alleviate some of the environmental conditions associated with them. All such studies rely on the existence of long-term consistent high-accuracy datasets of sea level. The Climate Change Initiative (CCI) of the European Space Agency was established in 2010 to provide improved time series of some ECVs, including sea level, with the purpose of providing such data openly to all to enable the widest possible utilisation of such data. Now in its second phase, the Sea Level CCI project (SL_cci) merges data from nine different altimeter missions in a clear, consistent and well-documented manner, selecting the most appropriate satellite orbits and geophysical corrections in order to further reduce the error budget. This paper summarises the corrections required, the provenance of corrections and the evaluation of options that have been adopted for the recently released v2.0 dataset (https://doi.org/10.5270/esa-sea_level_cci-1993_2015-v_2.0-201612). This information enables scientists and other users to clearly understand which corrections have been applied and their effects on the sea level dataset. The overall result of these changes is that the rate of rise of global mean sea level (GMSL) still equates to ∼ 3.2 mm yr−1 during 1992–2015, but there is now greater confidence in this result as the errors associated with several of the corrections have been reduced. Compared with v1.1 of the SL_cci dataset, the new rate of change is 0.2 mm yr−1 less during 1993 to 2001 and 0.2 mm yr−1 higher during 2002 to 2014. Application of new correction models brought a reduction of altimeter crossover variances for most corrections
Using lipidomics to reveal details of lipid accumulation in developing seeds from oilseed rape (Brassica napus L.)
With dwindling available agricultural land, concurrent with increased demand for oil, there is much current interest in raising oil crop productivity. We have been addressing this issue by studying the regulation of oil accumulation in oilseed rape (Brassica napus L). As part of this research we have carried out a detailed lipidomic analysis of developing seeds. The molecular species distribution in individual lipid classes revealed quite distinct patterns and showed where metabolic connections were important. As the seeds developed, the molecular species distributions changed, especially in the period of early (20 days after flowering, DAF) to mid phase (27DAF) of oil accumulation. The patterns of molecular species of diacylglycerol, phosphatidylcholine and acyl-CoAs were used to predict the possible relative contributions of diacylglycerol acyltransferase (DGAT) and phospholipid:diacylglycerol acyltransferase to triacylglycerol production. Our calculations suggest that DGAT may hold a more important role in influencing the molecular composition of TAG. Enzyme selectivity had an important influence on the final molecular species patterns. Our data contribute significantly to our understanding of lipid accumulation in the world’s third most important oil crop
On the validity of mean-field amplitude equations for counterpropagating wavetrains
We rigorously establish the validity of the equations describing the
evolution of one-dimensional long wavelength modulations of counterpropagating
wavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We
consider both periodic amplitude functions and localized wavepackets. For the
localized case, the wavetrains are completely decoupled at leading order, while
in the periodic case the amplitude equations take the form of mean-field
(nonlocal) Schr\"odinger equations rather than locally coupled partial
differential equations. The origin of this weakened coupling is traced to a
hidden translation symmetry in the linear problem, which is related to the
existence of a characteristic frame traveling at the group velocity of each
wavetrain. It is proved that solutions to the amplitude equations dominate the
dynamics of the governing equations on asymptotically long time scales. While
the details of the discussion are restricted to the class of model equations
having a leading cubic nonlinearity, the results strongly indicate that
mean-field evolution equations are generic for bimodal disturbances in
dispersive systems with \O(1) group velocity.Comment: 16 pages, uuencoded, tar-compressed Postscript fil
On the spectral properties of L_{+-} in three dimensions
This paper is part of the radial asymptotic stability analysis of the ground
state soliton for either the cubic nonlinear Schrodinger or Klein-Gordon
equations in three dimensions. We demonstrate by a rigorous method that the
linearized scalar operators which arise in this setting, traditionally denoted
by L_{+-}, satisfy the gap property, at least over the radial functions. This
means that the interval (0,1] does not contain any eigenvalues of L_{+-} and
that the threshold 1 is neither an eigenvalue nor a resonance. The gap property
is required in order to prove scattering to the ground states for solutions
starting on the center-stable manifold associated with these states. This paper
therefore provides the final installment in the proof of this scattering
property for the cubic Klein-Gordon and Schrodinger equations in the radial
case, see the recent theory of Nakanishi and the third author, as well as the
earlier work of the third author and Beceanu on NLS. The method developed here
is quite general, and applicable to other spectral problems which arise in the
theory of nonlinear equations
Selection of the ground state for nonlinear Schroedinger equations
We prove for a class of nonlinear Schr\"odinger systems (NLS) having two
nonlinear bound states that the (generic) large time behavior is characterized
by decay of the excited state, asymptotic approach to the nonlinear ground
state and dispersive radiation. Our analysis elucidates the mechanism through
which initial conditions which are very near the excited state branch evolve
into a (nonlinear) ground state, a phenomenon known as {\it ground state
selection}.
Key steps in the analysis are the introduction of a particular linearization
and the derivation of a normal form which reflects the dynamics on all time
scales and yields, in particular, nonlinear Master equations.
Then, a novel multiple time scale dynamic stability theory is developed.
Consequently, we give a detailed description of the asymptotic behavior of the
two bound state NLS for all small initial data. The methods are general and can
be extended to treat NLS with more than two bound states and more general
nonlinearities including those of Hartree-Fock type.Comment: Revision of 2001 preprint; 108 pages Te
Semiclassical Propagation of Coherent States for the Hartree equation
In this paper we consider the nonlinear Hartree equation in presence of a
given external potential, for an initial coherent state. Under suitable
smoothness assumptions, we approximate the solution in terms of a time
dependent coherent state, whose phase and amplitude can be determined by a
classical flow. The error can be estimated in by C \sqrt {\var}, \var
being the Planck constant. Finally we present a full formal asymptotic
expansion
Stable standing waves for a class of nonlinear Schroedinger-Poisson equations
We prove the existence of orbitally stable standing waves with prescribed
-norm for the following Schr\"odinger-Poisson type equation \label{intro}
%{%{ll} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0
\text{in} \R^{3}, %-\Delta\phi= |\psi|^{2}& \text{in} \R^{3},%. when . In the case we prove the existence and
stability only for sufficiently large -norm. In case our approach
recovers the result of Sanchez and Soler \cite{SS} %concerning the existence
and stability for sufficiently small charges. The main point is the analysis of
the compactness of minimizing sequences for the related constrained
minimization problem. In a final section a further application to the
Schr\"odinger equation involving the biharmonic operator is given
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