Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type

We prove local and global well-posedness for semi-relativistic, nonlinear Schr\"odinger equations $i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)$ with initial data in $H^s(\mathbb{R}^3)$, $s \geq 1/2$. Here $F(u)$ is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing $F(u)$, 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

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

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

Derivation of the Cubic Non-linear Schr\"odinger Equation from Quantum Dynamics of Many-Body Systems

We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schr\"odinger equation in a suitable scaling limit. The result is extended to $k$-particle density matrices for all positive integer $k$.Comment: 72 pages, 17 figures. Final versio

Sea level budget over 2003-2008: A reevaluation from GRACE space gravimetry, satellite altimetry and Argo

0921-8181From the IPCC 4th Assessment Report published in 2007, ocean thermal expansion contributed by similar to 50% to the 3.1 mm/yr observed global mean sea level rise during the 1993-2003 decade, the remaining rate of rise being essentially explained by shrinking of land ice. Recently published results suggest that since about 2003, ocean thermal expansion change, based on the newly deployed Argo system, is showing a plateau while sea level is still rising, although at a reduced rate (similar to 2.5 mm/yr). Using space gravimetry observations from GRACE, we show that recent years sea level rise can be mostly explained by an increase of the mass of the oceans. Estimating GRACE-based ice sheet mass balance and using published estimates for glaciers melting, we further show that ocean mass increase since 2003 results by about half from an enhanced contribution of the polar ice sheets - compared to the previous decade - and half from mountain glaciers melting. Taking also into account the small GRACE-based contribution from continental waters (<0.2 mm/yr), we find a total ocean mass contribution of similar to 2 mm/yr over 2003-2008. Such a value represents similar to 80% of the altimetry-based rate of sea level rise over that period. We next estimate the steric sea level (i.e., ocean thermal expansion plus salinity effects) contribution from: (1) the difference between altimetry-based sea level and ocean mass change and (2) Argo data. Inferred steric sea level rate from (1) (similar to 0.3 mm/yr over 2003-2008) agrees well with the Argo-based value also estimated here (0.37 mm/yr over 2004-2008). Furthermore, the sea level budget approach presented in this study allows us to constrain independent estimates of the Glacial Isostatic Adjustment (GIA) correction applied to GRACE-based ocean and ice sheet mass changes, as well as of glaciers melting. Values for the CIA correction and glacier contribution needed to close the sea level budget and explain GRACE-based mass estimates over the recent years agree well with totally independent determinations. (C) 2008 Elsevier B.V. All rights reserved

Collapse in the nonlocal nonlinear Schr\"odinger equation

We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schr\"{o}dinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both, Lyapunoff's method and virial identities. We find that for for a one-dimensional case, i.e. for $n=1$, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension $n\geq2$ and singular kernel $\sim 1/r^\alpha$, no collapse takes place if $\alpha<2$, whereas collapse is possible if $\alpha\ge2$. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of $\sim 1/r^2$ kernels. Moreover, different evolution scenarios for the three dimensional physically relevant case of Bose Einstein condensate are studied numerically for both, the ground state and a higher order toroidal state with and without an additional local repulsive nonlinear interaction. In particular, we show that presence of an additional local repulsive term can prevent collapse in those cases

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

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 $L^2$ by C \sqrt {\var}, \var being the Planck constant. Finally we present a full formal asymptotic expansion

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 stability of standing waves of Klein-Gordon equations in a semiclassical regime

We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.Comment: 9 page