172 research outputs found
Integrable lattice equations with vertex and bond variables
We present integrable lattice equations on a two dimensional square lattice
with coupled vertex and bond variables. In some of the models the vertex
dynamics is independent of the evolution of the bond variables, and one can
write the equations as non-autonomous "Yang-Baxter maps". We also present a
model in which the vertex and bond variables are fully coupled. Integrability
is tested with algebraic entropy as well as multidimensional consistencyComment: 15 pages, remarks added, other minor change
A modeling approach to assess the hydrological response of small mediterranean catchments to the variability of soil characteristics in a context of extreme events
This paper presents a modeling study aiming at quantifying the possible impact of soil characteristics on the hydrological response of small ungauged catchments in a context of extreme events. The study focuses on the September 2002 event in the Gard region (South-Eastern France), which led to catastrophic flash-floods. The proposed modeling approach is able to take into account rainfall variability and soil profiles variability. Its spatial discretization is determined using Digital Elevation Model (DEM) and a soil map. The model computes infiltration, ponding and vertical soil water distribution, as well as river discharge. In order to be applicable to ungauged catchments, the model is set up without any calibration and the soil parameter specification is based on an existing soil database. The model verification is based on a regional evaluation using 17 estimated discharges obtained from an extensive post-flood investigation. Thus, this approach provides a spatial view of the hydrological response across a large range of scales. To perform the simulations, radar rainfall estimations are used at a 1 km<sup>2</sup> and 5 min resolution. To specify the soil hydraulic properties, two types of pedotransfer function (PTF) are compared. It is shown that the PTF including information about soil structure reflects better the spatial variability that can be encountered in the field. The study is focused on four small ungauged catchments of less than 10 km<sup>2</sup>, which experienced casualties. Simulated specific peak discharges are found to be in agreement with estimations from a post-event in situ investigation. Examining the dynamics of simulated infiltration and saturation degrees, two different behaviors are shown which correspond to different runoff production mechanisms that could be encountered within catchments of less than 10 km<sup>2</sup>. They produce simulated runoff coefficients that evolve in time and highlight the variability of the infiltration capacity of the various soil types. Therefore, we propose a cartography distinguishing between areas prone to saturation excess and areas prone only to infiltration excess mechanisms. The questions raised by this modeling study will be useful to improve field observations, aiming at better understanding runoff generation for these extreme events and examine the possibility for early warning, even in very small ungauged catchments
On the Symmetries of Integrability
We show that the Yang-Baxter equations for two dimensional models admit as a
group of symmetry the infinite discrete group . The existence of
this symmetry explains the presence of a spectral parameter in the solutions of
the equations. We show that similarly, for three-dimensional vertex models and
the associated tetrahedron equations, there also exists an infinite discrete
group of symmetry. Although generalizing naturally the previous one, it is a
much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to
resolve the Yang-Baxter equations and their higher-dimensional generalizations
and initiate the study of three-dimensional vertex models. These symmetries are
naturally represented as birational projective transformations. They may
preserve non trivial algebraic varieties, and lead to proper parametrizations
of the models, be they integrable or not. We mention the relation existing
between spin models and the Bose-Messner algebras of algebraic combinatorics.
Our results also yield the generalization of the condition so often
mentioned in the theory of quantum groups, when no parameter is available.Comment: 23 page
A classification of four-state spin edge Potts models
We classify four-state spin models with interactions along the edges
according to their behavior under a specific group of symmetry transformations.
This analysis uses the measure of complexity of the action of the symmetries,
in the spirit of the study of discrete dynamical systems on the space of
parameters of the models, and aims at uncovering solvable ones. We find that
the action of these symmetries has low complexity (polynomial growth, zero
entropy). We obtain natural parametrizations of various models, among which an
unexpected elliptic parametrization of the four-state chiral Potts model, which
we use to localize possible integrability conditions associated with high genus
curves.Comment: 5 figure
On the complexity of some birational transformations
Using three different approaches, we analyze the complexity of various
birational maps constructed from simple operations (inversions) on square
matrices of arbitrary size. The first approach consists in the study of the
images of lines, and relies mainly on univariate polynomial algebra, the second
approach is a singularity analysis, and the third method is more numerical,
using integer arithmetics. Each method has its own domain of application, but
they give corroborating results, and lead us to a conjecture on the complexity
of a class of maps constructed from matrix inversions
The thermal-viscous disk instability model in the AGN context
Accretion disks in AGN should be subject to the same type of instability as
in cataclysmic variables (CVs) or in low-mass X-ray binaries (LMXBs), which
leads to dwarf nova and soft X-ray transient outbursts. It has been suggested
that this thermal/viscous instability can account for the long term variability
of AGNs. We test this assertion by presenting a systematic study of the
application of the disk instability model (DIM) to AGNs. We are using the
adaptative grid numerical code we have developed in the context of CVs,
enabling us to fully resolve the radial structure of the disk. We show that,
because in AGN disks the Mach numbers are very large, the heating and cooling
fronts are so narrow that they cannot be resolved by the numerical codes that
have been used until now. In addition, these fronts propagate on time scales
much shorter than the viscous time. As a result, a sequence of heating and
cooling fronts propagate back and forth in the disk, leading only to small
variations of the accretion rate onto the black hole, with short quiescent
states occurring for very low mass transfer rates only. Truncation of the inner
part of the disk by e.g. an ADAF does not alter this result, but enables longer
quiescent states. Finally we discuss the effects of irradiation by the central
X-ray source, and show that, even for extremely high irradiation efficiencies,
outbursts are not a natural outcome of the model.Comment: Astronomy & Astrophysics - in pres
Random Matrix Theory and higher genus integrability: the quantum chiral Potts model
We perform a Random Matrix Theory (RMT) analysis of the quantum four-state
chiral Potts chain for different sizes of the chain up to size L=8. Our
analysis gives clear evidence of a Gaussian Orthogonal Ensemble statistics,
suggesting the existence of a generalized time-reversal invariance.
Furthermore a change from the (generic) GOE distribution to a Poisson
distribution occurs when the integrability conditions are met. The chiral Potts
model is known to correspond to a (star-triangle) integrability associated with
curves of genus higher than zero or one. Therefore, the RMT analysis can also
be seen as a detector of ``higher genus integrability''.Comment: 23 pages and 10 figure
A birational mapping with a strange attractor: Post critical set and covariant curves
We consider some two-dimensional birational transformations. One of them is a
birational deformation of the H\'enon map. For some of these birational
mappings, the post critical set (i.e. the iterates of the critical set) is
infinite and we show that this gives straightforwardly the algebraic covariant
curves of the transformation when they exist. These covariant curves are used
to build the preserved meromorphic two-form. One may have also an infinite post
critical set yielding a covariant curve which is not algebraic (transcendent).
For two of the birational mappings considered, the post critical set is not
infinite and we claim that there is no algebraic covariant curve and no
preserved meromorphic two-form. For these two mappings with non infinite post
critical sets, attracting sets occur and we show that they pass the usual tests
(Lyapunov exponents and the fractal dimension) for being strange attractors.
The strange attractor of one of these two mappings is unbounded.Comment: 26 pages, 11 figure
Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm
The discrete-time Toda equation arises as a universal equation for the
relevant Hankel determinants associated with one-variable orthogonal
polynomials through the mechanism of adjacency, which amounts to the inclusion
of shifted weight functions in the orthogonality condition. In this paper we
extend this mechanism to a new class of two-variable orthogonal polynomials
where the variables are related via an elliptic curve. This leads to a `Higher
order Analogue of the Discrete-time Toda' (HADT) equation for the associated
Hankel determinants, together with its Lax pair, which is derived from the
relevant recurrence relations for the orthogonal polynomials. In a similar way
as the quotient-difference (QD) algorithm is related to the discrete-time Toda
equation, a novel quotient-quotient-difference (QQD) scheme is presented for
the HADT equation. We show that for both the HADT equation and the QQD scheme,
there exists well-posed -periodic initial value problems, for almost all
\s\in\Z^2. From the Lax-pairs we furthermore derive invariants for
corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page
Models of AM CVn star outbursts
Outbursting AM CVn stars exhibit outbursts similar to those observed in
different types of dwarf novae. Their light-curves combine the characteristic
features of SU UMa, ER UMa, Z Cam, and WZ Sge-type systems but also show a
variety of properties never observed in dwarf novae. The compactness of AM CVn
orbits and their unusual chemical composition make these systems valuable
testbeds for outburst models. We aim for a better understanding of the role of
helium in the accretion disc instability mechanism, testing the model for dwarf
novae outbursts in the case of AM CVn stars, and aim to explain the outburst
light-curves of these ultra-compact binaries. We calculated the properties of
the hydrogen-free AM CVn stars using our previously developed numerical code
adapted to the different chemical composition of these systems and supplemented
with formulae accounting for mass transfer rate variations, additional sources
of the disc heating, and the primary's magnetic field. We discovered how
helium-dominated discs react to the thermal-viscous instability and were able
to reproduce various features of the outburst cycles in the light-curves of AM
CVn stars. The AM CVn outbursts can be explained by the suitably adapted
dwarf-nova disc instability model but, as in the case of its application to
hydrogen-dominated cataclysmic variables, one has to resort to additional
mechanisms to account for the observed superoutbursts, dips, cycling states,
and standstills. We show that the enhanced mass-transfer rate, due presumably
to variable irradiation of the secondary, must not only be taken into account
but is a determining factor that shapes AM CVn star outbursts. The cause of the
variable secondary's irradiation has yet to be understood; the best candidate
is the precession of a tilted/warped disc.Comment: Astronomy and Astrophysics - in press; corrected (language) versio
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