Continuous symmetries of Lagrangians and exact solutions of discrete equations

One of the difficulties encountered when studying physical theories in discrete space-time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an invariant Lagrangian formalism for scalar single-variable difference schemes. The formalism is used to obtain first integrals and explicit exact solutions of the schemes. Equations invariant under two- and three- dimensional groups of Lagrangian symmetries are considered.Comment: 31 pages, submitted to Journal of Mathematical Physic

On the Linearization of Second-Order Differential and Difference Equations

This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and Noether-type integration technique. It turned out that there exist nonlinear superposition principles for solutions of special second-order ordinary difference equations which possess Lie group symmetries. This superposition springs from the linearization of second-order ordinary difference equations by means of non-point transformations which act simultaneously on equations and meshes. These transformations become some sort of contact transformations in the continuous limit.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

A heat transfer with a source: the complete set of invariant difference schemes

In this letter we present the set of invariant difference equations and meshes which preserve the Lie group symmetries of the equation u_{t}=(K(u)u_{x})_{x}+Q(u). All special cases of K(u) and Q(u) that extend the symmetry group admitted by the differential equation are considered. This paper completes the paper [J. Phys. A: Math. Gen. 30, no. 23 (1997) 8139-8155], where a few invariant models for heat transfer equations were presented.Comment: arxiv version is already officia

Symmetry-preserving discrete schemes for some heat transfer equations

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.Comment: 21 pages, 4 ps figure

An axisymmetric hydrodynamical model for the torus wind in AGN. III: Spectra from 3D radiation transfer calculations

We calculate a series of synthetic X-ray spectra from outflows originating from the obscuring torus in active galactic nuclei (AGN). Such modeling includes 2.5D hydrodynamical simulations of an X-ray excited torus wind, including the effects of X-ray heating, ionization, and radiation pressure. 3D radiation transfer calculations are performed in the 3D Sobolev approximation. Synthetic X-ray line spectra and individual profiles of several strong lines are shown at different inclination angles, observing times, and for different characteristics of the torus. Our calculations show that rich synthetic warm absorber spectra from 3D modeling are typically observed at a larger range of inclinations than was previously inferred from simple analysis of the transmitted spectra. In general, our results are supportive of warm absorber models based on the hypothesis of an "X-ray excited funnel flow" and are consistent with characteristics of such flows inferred from observations of warm absorbers from Seyfert 1 galaxies.Comment: 31 pages, 10 figure

AGN obscuration from winds: from dusty infrared-driven to warm and X-ray photoionized

We present calculations of AGN winds at ~parsec scales, along with the associated obscuration. We take into account the pressure of infrared radiation on dust grains and the interaction of X-rays from a central black hole with hot and cold plasma. Infrared radiation (IR) is incorporated in radiation-hydrodynamic simulations adopting the flux-limited diffusion approximation. We find that in the range of X-ray luminosities L=0.05 - 0.6 L_edd, the Compton-thick part of the flow (aka torus) has an opening angle of approximately 72-75 degrees regardless of the luminosity. At L > 0.1 L_edd the outflowing dusty wind provides the obscuration with IR pressure playing a major role. The global flow consists of two phases: the cold flow at inclinations \theta > 70 degrees and a hot, ionized wind of lower density at lower inclinations. The dynamical pressure of the hot wind is important in shaping the denser IR supported flow. At luminosities <0.1 L_edd episodes of outflow are followed by extended periods when the wind switches to slow accretion.Comment: accepted for publication in Ap

Active galaxy unification in the era of X-ray polarimetry

Active Galactic Nuclei (AGN), Seyfert galaxies and quasars, are powered by luminous accretion and often accompanied by winds which are powerful enough to affect the AGN mass budget, and whose observational appearance bears an imprint of processes which are happening within the central parsec around the black hole (BH). One example of such a wind is the partially ionized gas responsible for X-ray and UV absorption ('warm absorbers'). Here we show that such gas will have a distinct signature when viewed in polarized X-rays. Observations of such polarization can test models for the geometry of the flow, and the gas responsible for launching and collimating it. We present calculations which show that the polarization depends on the hydrodynamics of the flow, the quantum mechanics of resonance line scattering and the transfer of polarized X-ray light in the highly ionized moving gas. The results emphasize the three dimensional nature of the wind for modeling spectra. We show that the polarization in the 0.1-10 keV energy range is dominated by the effects of resonance lines. We predict a $5-25$ X-ray polarization signature of type-2 objects in this energy range. These results are general to flows which originate from a cold torus-like structure, located $\sim 1$pc from the BH, which wraps the BH and is ultimately responsible for the apparent dichotomy between type 1 and type 2 AGNs. Such signals will be detectable by future dedicated X-ray polarimetry space missions, such as the NASA Gravity and Extreme Magnetism SMEX, GEMS.Comment: 13 pages, 4 figures, ApJ Letters accepted for publicatio