Travelling waves for the Gross-Pitaevskii equation II

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results, where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.Comment: Final version accepted for publication in Communications in Mathematical Physics with a few minor corrections and added remark

Minimal energy for the traveling waves of the Landau-Lifshitz equation

We consider nontrivial finite energy traveling waves for the Landau-Lifshitz equation with easy-plane anisotropy. Our main result is the existence of a minimal energy for these traveling waves, in dimensions two, three and four. The proof relies on a priori estimates related with the theory of harmonic maps and the connection of the Landau-Lifshitz equation with the kernels appearing in the Gross-Pitaevskii equation.Comment: submitte

Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields

Using the Pauli-Villars regularization and arguments from convex analysis, we construct solutions to the classical time-independent Maxwell equations in Dirac's vacuum, in the presence of small external electromagnetic sources. The vacuum is not an empty space, but rather a quantum fluctuating medium which behaves as a nonlinear polarizable material. Its behavior is described by a Dirac equation involving infinitely many particles. The quantum corrections to the usual Maxwell equations are nonlinear and nonlocal. Even if photons are described by a purely classical electromagnetic field, the resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics.Comment: Final version to appear in Arch. Rat. Mech. Analysi

Renormalization and asymptotic expansion of Dirac's polarized vacuum

We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no `real' electron. We show that it admits an asymptotic expansion to any order in powers of the physical coupling constant \alphaph, provided that the ultraviolet cut-off behaves as \Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1. The renormalization parameter $

Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms

We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant $\alpha$. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an $L^1$ function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to $\alpha$ by the formula $\alpha_{\rm phys}\simeq \alpha(1+2\alpha/(3\pi)\log\Lambda)^{-1}$, where $\Lambda$ is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge $Z$. In the nonrelativistic limit, we obtain that this number is $\leq 2Z$, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Sere and Solovej on the mean-field approximation of no-photon QED.Comment: 37 pages, 1 figur

Quantitative analysis of infrared spectra of adsorbed species using transmission and diffuse reflectance modes Case study: Heats of adsorption of CO on TiO2 and CuO/Al2O3

Couble, Julien Gravejat, Paul Gaillard, Francois Bianchi, DanielIR spectroscopy in transmission and in diffuse reflectance modes is used to characterize the adsorbed CO species formed during the adsorption of CO on (a) a TiO2 solid and (b) a partially reduced CuO/Al2O3 solid. The experiments consist studying the evolution of the intensities of the characteristic IR bands with the adsorption temperature T-a at a constant adsorption pressure P-a. For the two IR analytical procedures, the spectra are used to measure the heats of adsorption of the adsorbed CO species on the two solids according to the AEIR procedure previously described using the IR transmission mode. This allows us to fix the experimental conditions allowing a quantitative analysis of the IR spectra recorded in diffuse reflectance. Mainly, it is shown that the relative reflectance at the position of the characteristic IR bands of the adsorbed CO species must be higher than a critical value during the measurements. According to the amount of adsorbed species, this can be achieved by the dilution of the catalyst with another material that does not contribute to the experimental data. In the present study, it is shown that TiO2 and CuO/Al2O3 can be used without and with dilution respectively. For the Cu/Al2O3 catalyst, the alumina support has been used for the dilution whereas KBr that exhibits acceptable optical properties leads to a significant modification of the heats of adsorption of the adsorbed CO species. This study suggests that heats of adsorption of adsorbed species formed on catalysts of low IR transmission can be determined by the AEIR method using the diffuse reflectance mode. This offers a way to extend the application of quantitative analysis of IR spectra of adsorbed species (i.e. AEIR method) to solid of low IR transmission in particular the industrial catalysts. (C) 2009 Elsevier B.V. All rights reserved

Impact of potassium on the heats of adsorption of adsorbed CO species on supported Pt particles by using the AEIR method

International audienceThe heats of adsorption at several coverages of the linear and bridged CO species (denoted L and B, respectively) adsorbed on the Pt0 sites of the 2.9 wt% Pt/10% K/Al2O3 catalyst are determined using the Adsorption Equilibrium Infrared spectroscopy method. The addition of K on 2.9% Pt/Al2O3 modifies significantly the adsorption of CO on the Pt particles: (a) the ratio L/B is decreased from 8.4 to 1, (b) a new adsorbed CO species is detected with an IR band at 1763 cm−1, (c) the heats of adsorption of L and B CO species are significantly altered and the positions of their IR bands are shifted. The heats of adsorption of L CO species are decreased: i.e. 206 and 105 kJ/mol at low coverages on Pt/Al2O3 and Pt/K/Al2O3 respectively. Two B CO species denoted B1 and B2, with different heats of adsorption are observed on Pt/K/Al2O3. The heats of adsorption of B2 CO species (major B CO species) are significantly larger than those measured in the absence of K: i.e. 94 and 160 kJ/mol at low coverages on Pt/Al2O3 and Pt/K/Al2O3 respectively, whereas those of B1 CO species (minor species) are similar: 90 kJ/mol at low coverages. These values are consistent with the qualitative High Resolution Electron Energy Loss Spectrometry literature data on Pt(1 1 1) modified by potassium

Nonexistence of supersonic traveling waves for nonlinear Schrödinger equations with non-zero conditions at infinity

International audienc

Heats of Adsorption of Linear and Bridged CO Species Adsorbed on a 3% Ag/Al2O3 Catalyst Using in situ FTIR Spectroscopy under Adsorption Equilibrium

International audienceThe heats of adsorption of linear and bridged CO species adsorbed on Ag0 sites of a reduced 3% Ag/Al2O3 catalyst and of the linear CO species on Ag+ sites of the oxidized catalyst are determined as the function of their respective coverages by using the adsorption equilibrium infrared spectroscopy method previously developed. The evolutions of the intensities of the IR bands of each adsorbed species: 2045 cm-1 and 2000 cm-1 at 300 K, for the linear and bridged CO species, respectively, on Ag0 sites of the reduced solid and 2168 cm-1 for a linear CO species on Ag+ sites of the oxidized solid are determined as a function of the adsorption temperature Ta, at a constant CO adsorption pressure PCO. This provides the evolutions of the coverages of each adsorbed CO species with Ta in isobar conditions that give the individual heats of adsorption of the three adsorbed CO species at several coverages according to an adsorption model. The heats of adsorption of the linear and bridged adsorbed CO species on Ag0 sites vary linearly with their coverages : from E0 = 76 kJ/mol to E1 = 58 kJ/mol for the linear CO species and from E0 = 88 kJ/mol to E1 = 84 kJ/mol for the bridged CO species at coverages 0 and 1, respectively. Volumetric measurements indicate that the total amount of the adsorbed CO species represents a small fraction of the superficial sites of the reduced Ag particles suggesting that they are adsorbed on defect sites. The heat of adsorption of the linear CO species on the Ag+ sites linearly varies with its coverage from E0 = 66 kJ/mol to E1 = 41 kJ/mol at coverages 0 and 1, respectively