Dispersion and collapse of wave maps

We study numerically the Cauchy problem for equivariant wave maps from 3+1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behavior in formation of black holes.Comment: 14 pages, Latex, 9 eps figures included, typos correcte

A deep Chandra observation of the Perseus cluster: shocks and ripples

We present preliminary results from a deep observation lasting almost 200 ks, of the centre of the Perseus cluster of galaxies around NGC 1275. The X-ray surface brightness of the intracluster gas beyond the inner 20 kpc, which contains the inner radio bubbles, is very smooth apart from some low amplitude quasi-periodic ripples. A clear density jump at a radius of 24 kpc to the NE, about 10 kpc out from the bubble rim, appears to be due to a weak shock driven by the northern radio bubble. A similar front may exist round both inner bubbles but is masked elsewhere by rim emission from bright cooler gas. The continuous blowing of bubbles by the central radio source, leading to the propagation of weak shocks and viscously-dissipating sound waves seen as the observed fronts and ripples, gives a rate of working which balances the radiative cooling within the inner 50 kpc of the cluster core.Comment: Accepted for publication in MNRAS (minor changes) Higher picture quality available from http://www-xray.ast.cam.ac.uk/papers/per_200ks.pd

Star Formation, Radio Sources, Cooling X-ray Gas, and Galaxy Interactions in the Brightest Cluster Galaxy in 2A0335+096

We present deep emission-line imaging taken with the SOAR Optical Imaging Camera of the brightest cluster galaxy (BCG) in the nearby (z=0.035) X-ray cluster 2A0335+096. We analyze long-slit optical spectroscopy, archival VLA, Chandra X-ray, and XMM UV data. 2A0335+096 is a bright, cool-core X-ray cluster, once known as a cooling flow. Within the highly disturbed core revealed by Chandra X-ray observations, 2A0335+096 hosts a highly structured optical emission-line system. The redshift of the companion is within 100 km/s of the BCG and has certainly interacted with the BCG, and is likely bound to it. The comparison of optical and radio images shows curved filaments in H-alpha emission surrounding the resolved radio source. The velocity structure of the emission-line bar between the BCG nucleus and the companion galaxy provides strong evidence for an interaction between the two in the last ~50 Myrs. The age of the radio source is similar to the interaction time, so this interaction may have provoked an episode of radio activity. We estimate a star formation rate of >7 solar mass/yr based on the Halpha and archival UV data, a rate similar to, but somewhat lower than, the revised X-ray cooling rate of 10-30 solar masses/year estimated from XMM spectra by Peterson & workers. The Halpha nebula is limited to a region of high X-ray surface brightness and cool X-ray temperature. The detailed structures of H-alpha and X-ray gas differ. The peak of the X-ray emission is not the peak of H-alpha emission, nor does it lie in the BCG. The estimated age of the radio lobes and their interaction with the optical emission-line gas, the estimated timescale for depletion and accumulation of cold gas, and the dynamical time in the system are all similar, suggesting a common trigger mechanism.Comment: Accepted AJ, July 2007 publication. Vol 134, p. 14-2

Chaotic Orbits in Thermal-Equilibrium Beams: Existence and Dynamical Implications

Phase mixing of chaotic orbits exponentially distributes these orbits through their accessible phase space. This phenomenon, commonly called ``chaotic mixing'', stands in marked contrast to phase mixing of regular orbits which proceeds as a power law in time. It is operationally irreversible; hence, its associated e-folding time scale sets a condition on any process envisioned for emittance compensation. A key question is whether beams can support chaotic orbits, and if so, under what conditions? We numerically investigate the parameter space of three-dimensional thermal-equilibrium beams with space charge, confined by linear external focusing forces, to determine whether the associated potentials support chaotic orbits. We find that a large subset of the parameter space does support chaos and, in turn, chaotic mixing. Details and implications are enumerated.Comment: 39 pages, including 14 figure

Asymptotic Spectroscopy of Rotating Black Holes

We calculate analytically the transmission and reflection amplitudes for waves incident on a rotating black hole in d=4, analytically continued to asymptotically large, nearly imaginary frequency. These amplitudes determine the asymptotic resonant frequencies of the black hole, including quasinormal modes, total-transmission modes and total-reflection modes. We identify these modes with semiclassical bound states of a one-dimensional Schrodinger equation, localized along contours in the complexified r-plane which connect turning points of corresponding null geodesics. Each family of modes has a characteristic temperature and chemical potential. The relations between them provide hints about the microscopic description of the black hole in this asymptotic regime.Comment: References adde

Analytic Behaviour of Competition among Three Species

We analyse the classical model of competition between three species studied by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the approaches of singularity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations between critical values arising from the approach of dynamical systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic