Cauchy problem and quasi-stationary limit for the Maxwell-Landau-Lifschitz and Maxwell-Bloch equations

In this paper we continue the investigation of the Maxwell-Landau-Lifschitz and Maxwell-Bloch equations. In particular we extend some previous results about the Cauchy problem and the quasi-stationary limit to the case where the magnetic permeability and the electric permittivity are variable

Justification of the log-KdV equation in granular chains: the case of precompression

For travelling waves with nonzero boundary conditions, we justify the logarithmic Korteweg-de Vries equation as the leading approximation of the Fermi-Pasta-Ulam lattice with Hertzian nonlinear potential in the limit of small anharmonicity. We prove control of the approximation error for the travelling solutions satisfying differential advance-delay equations, as well as control of the approximation error for time-dependent solutions to the lattice equations on long but finite time intervals. We also show nonlinear stability of the travelling waves on long but finite time intervals.Comment: 29 page

On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations

In this paper we deal with weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall-Magneto-Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.Comment: 45 page

Geometric optics and instability for NLS and Davey-Stewartson models

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrodinger equations with gauge invariant power-law nonlinearities and non-local perturbations. The model includes the Davey--Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variant. Our analysis reveals a new localization phenomenon for non-local perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach.Comment: 33 page

From Newton's cradle to the discrete p-Schr\"odinger equation

We investigate the dynamics of a chain of oscillators coupled by fully-nonlinear interaction potentials. This class of models includes Newton's cradle with Hertzian contact interactions between neighbors. By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times. The envelope equation leading to approximate solutions is a discrete p-Schr\"odinger equation. Our results include the existence of long-lived breather solutions to the original model. For a large class of localized initial conditions, we also estimate the maximal decay of small amplitude solutions over long times

The Central Kiloparsec of Seyfert and Inactive Host Galaxies: a Comparison of Two-Dimensional Stellar and Gaseous Kinematics

We investigate the properties of the two-dimensional distribution and kinematics of ionised gas and stars in the central kiloparsecs of a matched sample of nearby active (Seyfert) and inactive galaxies, using the SAURON Integral Field Unit on the William Herschel Telescope. The ionised gas distributions show a range of low excitation regions such as star formation rings in Seyferts and inactive galaxies, and high excitation regions related to photoionisation by the AGN. The stellar kinematics of all galaxies in the sample show regular rotation patterns typical of disc-like systems, with kinematic axes which are well aligned with those derived from the outer photometry and which provide a reliable representation of the galactic line of nodes. After removal of the non-gravitational components due to e.g. AGN-driven outflows, the ionised gas kinematics in both the Seyfert and inactive galaxies are also dominated by rotation with global alignment between stars and gas in most galaxies. This result is consistent with previous findings from photometric studies that the large-scale light distribution of Seyfert hosts are similar to inactive hosts. However, fully exploiting the two-dimensional nature of our spectroscopic data, deviations from axisymmetric rotation in the gaseous velocity fields are identified that suggest the gaseous kinematics are more disturbed at small radii in the Seyfert galaxies compared with the inactive galaxies, providing a tentative link between nuclear gaseous streaming and nuclear activity.Comment: Accepted for publication in MNRAS, 34 pages, 20 figure