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$^{th}$ 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 $\mathbb{R}$. 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 $n$ spins coupled to a single harmonic oscillator. We focus on the singularities of the vector-valued moment map whose components are the $n+1$ 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