arXiv subjects

Alfred Ramani

Publications and source records attributed to Alfred Ramani.

28 records · Page 2Linked to original sources

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↗

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

Integrable lattice equations and their growth properties

In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree grows exponentially fast, for integrable lattice equations the degree growth is polynomial. The growth criterion is used in order to obtain the integrable deautonomisations of the equations examined. In the case of linearisable lattice equations we show that the degree growth is slower than in the case of equations integrable through Inverse Scattering Transform techniques.

math-ph↗

arXiv2007

Integrable systems without the Painlevé property

We examine whether the Painlevé property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlevé continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities.

math-ph↗

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↗