80,011 research outputs found
An Analytical Study on the Multi-critical Behaviour and Related Bifurcation Phenomena for Relativistic Black Hole Accretion
We apply the theory of algebraic polynomials to analytically study the
transonic properties of general relativistic hydrodynamic axisymmetric
accretion onto non-rotating astrophysical black holes. For such accretion
phenomena, the conserved specific energy of the flow, which turns out to be one
of the two first integrals of motion in the system studied, can be expressed as
a 8 degree polynomial of the critical point of the flow configuration.
We then construct the corresponding Sturm's chain algorithm to calculate the
number of real roots lying within the astrophysically relevant domain of
. This allows, for the first time in literature, to {\it
analytically} find out the maximum number of physically acceptable solution an
accretion flow with certain geometric configuration, space-time metric, and
equation of state can have, and thus to investigate its multi-critical
properties {\it completely analytically}, for accretion flow in which the
location of the critical points can not be computed without taking recourse to
the numerical scheme. This work can further be generalized to analytically
calculate the maximal number of equilibrium points certain autonomous dynamical
system can have in general. We also demonstrate how the transition from a
mono-critical to multi-critical (or vice versa) flow configuration can be
realized through the saddle-centre bifurcation phenomena using certain
techniques of the catastrophe theory.Comment: 19 pages, 2 eps figures, to appear in "General Relativity and
Gravitation
Multi-mode solitons in the classical Dicke-Jaynes-Cummings-Gaudin Model
We present a detailed analysis of the classical Dicke-Jaynes-Cummings-Gaudin
integrable model, which describes a system of spins coupled to a single
harmonic oscillator. We focus on the singularities of the vector-valued moment
map whose components are the mutually commuting conserved Hamiltonians.
The level sets of the moment map corresponding to singular values may be viewed
as degenerate and often singular Arnold-Liouville torii. A particularly
interesting example of singularity corresponds to unstable equilibrium points
where the rank of the moment map is zero, or singular lines where the rank is
one. The corresponding level sets can be described as a reunion of smooth
strata of various dimensions. Using the Lax representation, the associated
spectral curve and the separated variables, we show how to construct
explicitely these level sets. A main difficulty in this task is to select,
among possible complex solutions, the physically admissible family for which
all the spin components are real. We obtain explicit solutions to this problem
in the rank zero and one cases. Remarkably this corresponds exactly to
solutions obtained previously by Yuzbashyan and whose geometrical meaning is
therefore revealed. These solutions can be described as multi-mode solitons
which can live on strata of arbitrary large dimension. In these solitons, the
energy initially stored in some excited spins (or atoms) is transferred at
finite times to the oscillator mode (photon) and eventually comes back into the
spin subsystem. But their multi-mode character is reflected by a large
diversity in their shape, which is controlled by the choice of the initial
condition on the stratum
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