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 $A_2^{(1)}$. 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 $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ 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 $s$-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