arXiv subjects

Basil Grammaticos

Publications and source records attributed to Basil Grammaticos.

33 records · Page 2Linked to original sources

arXiv2017

Calculating the algebraic entropy of mappings with unconfined singularities

We present a method for calculating the dynamical degree of a mapping with unconfined singularities. It is based on a method introduced by Halburd for the computation of the growth of the iterates of a rational mapping with confined singularities. In particular, we show through several examples how simple calculations, based on the singularity patterns of the mapping, allow one to obtain the exact value of the dynamical degree for nonintegrable mappings that do not possess the singularity confinement property. We also study linearisable mappings with unconfined singularities to show that in this case our method indeed yields zero algebraic entropy.

math-ph↗

arXiv2017

Integrable mappings and the notion of anticonfinement

We examine the notion of anticonfinement and the role it has to play in the singularity analysis of discrete systems. A singularity is said to be anticonfined if singular values continue to arise indefinitely for the forward and backward iterations of a mapping, with only a finite number of iterates taking regular values in between. We show through several concrete examples that the behaviour of some anticonfined singularities is strongly related to the integrability properties of the discrete mappings in which they arise, and we explain how to use this information to decide on the integrability or non-integrability of the mapping.

math-ph↗

arXiv2017

A systematic method for constructing discrete Painlevé equations in the degeneration cascade of the E$_8$ group

We present a systematic and quite elementary method for constructing discrete Painlevé equations in the degeneration cascade for E$_8^{(1)}$. Starting from the invariant for the autonomous limit of the E$_8^{(1)}$ equation one wishes to study, the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms. These new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping. We explain the method on three examples, two symmetric mappings and an asymmetric one, and we discuss the link between our results and the known geometric structure of these mappings.

math-ph↗

arXiv2017

Multiplicative equations related to the affine Weyl group E$_8$

We derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated to the affine Weyl group E$_8^{(1)}$. Five such systems are obtained, three of which turn out to be linearisable and the remaining two are integrable in terms of elliptic functions. In the case of the linearisable mappings we derive nonautonomous forms which contain a free function of the dependent variable and we present the linearisation in each case. The two remaining systems are deautonomised to new discrete Painlevé equations. We show that these equations are in fact special forms of much richer systems associated to the affine Weyl groups E$_7^{(1)}$ and E$_8^{(1)}$ respectively.

math-ph↗

arXiv2016

Calculating algebraic entropies: an express method

We describe a method for investigating the integrable character of a given three-point mapping, provided that the mapping has confined singularities. Our method, dubbed "express", is inspired by a novel approach recently proposed by R.G. Halburd. While the latter aims at computing the exact degree growth of a given mapping based on the structure of its singularities, we content ourselves with obtaining an answer as to whether a given system is integrable or not. We present several examples illustrating our method as well as its limitations. We also compare the present method to the full-deautonomisation approach we recently introduced.

math-ph↗

arXiv2016

Full-deautonomisation of a lattice equation

In this letter we report on the unexpected possibility of applying the full-deautonomisation approach we recently proposed for predicting the algebraic entropy of second-order birational mappings, to discrete lattice equations. Moreover, we show, on two examples, that the full-deautonomisation technique can in fact also be successfully applied to reductions of these lattice equations to mappings with orders higher than 2. In particular, we apply this technique to a recently discovered lattice equation that has confined singularities while being nonintegrable, and we show that our approach accurately predicts this nonintegrable character. Finally, we demonstrate how our method can even be used to predict the algebraic entropy for some nonconfining higher order mappings.

math-ph↗

arXiv2015

Singularity confinement and full-deautonomisation: a discrete integrability criterion

We present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector. Our method is based on the full-deautonomisation procedure, which consists in analysing non-autonomous extensions of a given discrete system obtained by adding terms that are initially absent, but whose presence does not alter the singularity pattern. A justification for this approach is given through an algebro-geometric analysis. We also introduce the notions of early and late confinement. While the former is a confinement that may exist already for the autonomous system, the latter corresponds to a singularity pattern longer than that of the autonomous case. Late confinement will be shown to play an important role in the singularity analysis of systems with non-trivial gauge freedom, for which the existence of an undetected gauge in conjunction with a sketchy analysis, might lead to erroneous conclusions as to their integrability. An algebro-geometric analysis of the role of late confinement in this context is also offered. This novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping.

math-ph↗

arXiv2014

The redemption of singularity confinement

We present a novel way to apply the singularity confinement property as a discrete integrability criterion. We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as functions of the independent variable, applied to a mapping complemented with terms that are absent in the original mapping but which do not change the singularity structure. We shall show, on a host of examples including the well-known mapping of Hietarinta-Viallet, that our approach offers a way to compute the algebraic entropy for these mappings exactly, thereby allowing one to distinguish between the integrable and non-integrable cases even when both have confined singularities.

nlin.SI↗

arXiv2014

Deautonomisation by singularity confinement: an algebro-geometric justification

The `deautonomisation' of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage invariably leads to a nonintegrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painleve equations, how their regularisation through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.

nlin.SI↗

arXiv2011

Dynamical Studies of Equations from the Gambier Family

We consider the hierarchy of higher-order Riccati equations and establish their connection with the Gambier equation. Moreover we investigate the relation of equations of the Gambier family to other nonlinear differential systems. In particular we explore their connection to the generalized Ermakov-Pinney and Milne-Pinney equations. In addition we investigate the consequence of introducing Okamoto's folding transformation which maps the reduced Gambier equation to a Liénard type equation. Finally the conjugate Hamiltonian aspects of certain equations belonging to this family and their connection with superintegrability are explored.

nlin.SI↗

arXiv2009

Modeling tumor cell migration: from microscopic to macroscopic

It has been shown experimentally that contact interactions may influence the migration of cancer cells. Previous works have modelized this thanks to stochastic, discrete models (cellular automata) at the cell level. However, for the study of the growth of real-size tumors with several millions of cells, it is best to use a macroscopic model having the form of a partial differential equation (PDE) for the density of cells. The difficulty is to predict the effect, at the macroscopic scale, of contact interactions that take place at the microscopic scale. To address this we use a multiscale approach: starting from a very simple, yet experimentally validated, microscopic model of migration with contact interactions, we derive a macroscopic model. We show that a diffusion equation arises, as is often postulated in the field of glioma modeling, but it is nonlinear because of the interactions. We give the explicit dependence of diffusivity on the cell density and on a parameter governing cell-cell interactions. We discuss in details the conditions of validity of the approximations used in the derivation and we compare analytic results from our PDE to numerical simulations and to some in vitro experiments. We notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses, supercooled liquids and jamming systems.

q-bio.CB↗

arXiv2008

Quantum Painlevé Equations: from Continuous to Discrete

We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations.

math.QA↗

arXiv2007

Do All Integrable Evolution Equations Have the Painlevé Property?

We examine whether the Painleve property is necessary for the integrability of partial differential equations (PDEs). We show that in analogy to what happens in the case of ordinary differential equations (ODEs) there exists a class of PDEs, integrable through linearisation, which do not possess the Painleve property. The same question is addressed in a discrete setting where we show that there exist linearisable lattice equations which do not possess the singularity confinement property (again in analogy to the one-dimensional case).

nlin.SI↗

arXiv1997

Constructing Integrable Third Order Systems:The Gambier Approach

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlevé method for ODE's and the singularity confinement method for mappings).

solv-int↗

arXiv1993

Casorati Determinant Solutions for the Discrete Painlevé-II Equation

We present a class of solutions to the discrete Painlevé-II equation for particular values of its parameters. It is shown that these solutions can be expressed in terms of Casorati determinants whose entries are discrete Airy functions. The analogy between the $τ$ function for the discrete P$_{\rm \romanno2}$ and the that of the discrete Toda molecule equation is pointed out.

solv-int↗