arXiv subjects

Alfred Ramani

Publications and source records attributed to Alfred Ramani.

At least 19 recordsLinked to original sources

arXiv2026

Singularity interactions for lattice equations: introducing the taishi

We study the singularities of some selected integrable lattice equations and show they all admit three types of singularities: one of finite and two of infinite extent. In particular, we show that all the equations we study possess a recently discovered, strip-like, singularity which is known under the moniker of ''taishi''. We study in detail the interaction of these taishi with the remaining two types of singularities and show that the rich behaviour first obtained in the case of the Korteweg-deVries equation is also present for other equations. Moreover, we find that in all cases this behaviour can be encoded in terms of simple symbolic dynamics which are just the dynamics governing a Box & Ball cellular automaton.

nlin.SI↗

arXiv2026

Deautonomising the Lyness mapping

We examine the Lyness mapping (an integrable $N$th-order discrete system which can be generated from a one-dimensional reduction of the Hirota-Miwa equation) from the point of view of deautonomisation. We show that only the $N=2$ case can be deautonomised when one works with the standard form of the mapping. However it turns out that deautonomisation is possible for arbitrary $N$ when one considers the derivative form of the Lyness mapping. The deautonomisation of the derivative of the $N=2$ case leads to a result we have never met before: the secular dependence in the coefficients of the mapping enters through two different exponential terms instead of just a single one. As a consequence, it turns out that a limit of this multiplicative dependence towards an additive one is possible without modifying the dependent variable. Finally, the analysis of the `late' singularity confinement of the $N=2$ case leads to a novel realisation of the full-deautonomisation principle: the dynamical degree is not given (as is customary) simply by the solution of some linear or multiplicative equation, but is present in the growth of the non-linear (and non-integrable) late-confinement conditions.

nlin.SI↗

arXiv2026

The trouble with deautonomising higher order maps

The deautonomisation of birational maps that have the singularity confinement property, i.e. the construction of nonautonomous versions of such maps that preserve the singularity properties of the original, has proven crucial in our understanding of the mathematical properties behind the integrability of second order maps. For example, the deautonomisation procedure led directly to the development of a general theory of discrete Painlev\'e equations, and it seems highly likely it will play a crucial role in any future theory of higher dimensional Painlev\'e equations as well. Generally speaking however, higher order integrable mappings may have non-confined singularities and it is important to understand if, and how, deautonomisation should work for such mappings. In this paper we explore different deautonomisation scenarios on a series of carefully constructed higher order mappings, integrable as well as non-integrable, that possess non-confined singularities and we challenge some common assumptions regarding the co-dimensionality of the singular loci that might play a role in the deautonomisation process. Along the way we also propose a novel procedure to calculate the growth of the multiplicities of singularities that appear in so-called anticonfined singularity patterns, based on an ultradiscrete version of the mapping.

nlin.SI↗

arXiv2025

A fast algorithmic way to calculate the degree growth of birational mappings

We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings, based on Halburd's method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer arithmetic with additions and, in some cases, multiplications by small integers. It is therefore extremely fast. Several examples of integrable and non-integrable mappings are presented. In the latter case the dynamical degree we obtain from our method is always in agreement with that calculated by previously known methods.

nlin.SI↗

arXiv2024

Singularities and growth of higher order discrete equations

We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are integrable through spectral methods, typically belonging to the QRT family, to a variety of linearisable ones. We show that by judiciously choosing these linearisable mappings, it is possible to obtain higher order mappings that exhibit the maximal degree growth compatible with integrability, i.e. for which the degree grows as a polynomial of order equal to the order of the mapping. In all the cases we analysed, we found that maximal degree growth was associated with the existence of an unconfining singularity pattern. Several cases with submaximal growth but which still possess unconfining singularity patterns are also presented. In many cases the exact degrees of the iterates of the mappings were obtained by applying a method due to Halburd, based on the preimages of specific values that appear in the singularity patterns of the mapping, but we also present some examples where such a calculation appears to be impossible.

nlin.SI↗

arXiv2020

On the singularities of the discrete Korteweg-deVries equation

We study the structure of singularities in the discrete Korteweg-deVries (d-KdV) equation. Four different types of singularities are identified. The first type corresponds to localised, `confined', singularities, the confinement constraints for which provide the integrability conditions for generalisations of d-KdV. Two other types of singularities are of infinite extent and consist of oblique lines of infinities, possibly alternating with lines of zeros. The fourth type of singularity corresponds to horizontal strips where the product of the values on vertically adjacent points is equal to 1. (A vertical version of this singularity with product equal to $-1$ on horizontally adjacent sites also exists). Due to its orientation this singularity can, in fact, interact with the other types. This leads to an extremely rich structure for the singularities of d-KdV, which is studied in detail in this paper. Given the important role played by the fourth type of singularity we decided to give it a special name: taishi (the origin of which is explained in the text). The taishi do not exist for nonintegrable extensions of d-KdV, which explains the relative paucity of singularity structures in the nonintegrable case: the second and third type of singularities that correspond to oblique lines still exist and the localised singularities of the integrable case now become unconfined, leading to semi-infinite lines of infinities alternating with zeros.

math-ph↗

arXiv2019

On the singularity structure of the discrete KdV equation

The discrete KdV (dKdV) equation, the pinnacle of discrete integrability, is often thought to possess the singularity confinement property because it confines on an elementary quadrilateral. Here we investigate the singularity structure of the dKdV equation through reductions of the equation, obtained for initial conditions on a staircase with height 1, and show that it is much more subtle than one might assume. We first study the singularities for the mappings obtained after reduction and contrast these with the singularities that arise in non-integrable generalizations of those mappings. We then show that the so-called `express method' for obtaining dynamical degrees for second order mappings can be succesfully applied to all the higher order mappings we derived. Finally, we use the information obtained on the singularity structure of the reductions to describe an important subset of singularity patterns for the dKdV equation and we present an example of a non-confining pattern and explain why its existence does not contradict the integrability of the dKdV equation.

math-ph↗

arXiv2019

Discrete Painlev\'e equations from singularity patterns: the asymmetric trihomographic case

We derive the discrete Painlev\'e equations associated to the affine Weyl group E$_8^{(1)}$ that can be represented by an (in the QRT sense) "asymmetric" trihomographic system. The method used in this paper is based on singularity confinement. We start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns which cannot lead to confined singularities. Working with the remaining ones we implement the confinement conditions and derive the corresponding discrete Painlev\'e equations, which involve two variables. By eliminating either of these variables we obtain a "symmetric" equation. Examining all these equations of a single variable, we find that they coincide exactly with those derived in previous works of ours, thereby establishing the completeness of our results.

math-ph↗

arXiv2019

Singularity confinement as an integrability criterion

In this paper we present a rigorous method for deciding whether a birational three point mapping that has the singularity confinement property is integrable or not, based only on the structure of its (confined) singularity patterns. We also explain how the exact value of the dynamical degree for such a mapping may be deduced from the singularity patterns.

math-ph↗

arXiv2019

Constructing discrete Painlevé equations: from E$_8^{(1)}$ to A$_1^{(1)}$ and back

The `restoration method' is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with E$_8^{(1)}$ symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that are equivalent to it by homographic transformations. Discrete Painlevé equations are then obtained by deautonomising the various mappings thus obtained. We apply the restoration method to two challenging examples, one of which does not lead to a QRT mapping at the autonomous limit but we verify that even in that case our method is indeed still applicable. For one of the equations we derive we also show how, starting from a form where the independent variable advances one step at a time, we can obtain versions that correspond to multiple-step evolutions.

math-ph↗

arXiv2018

Restoring discrete Painlev\'e equations from an E$_8^{(1)}$-associated one

We present a systematic method for the construction of discrete Painlev\'e equations. The method, dubbed `restoration', allows one to obtain all discrete Painlev\'e equations that share a common autonomous limit, up to homographic transformations, starting from any one of those limits. As the restoration process crucially depends on the classification of canonical forms for the mappings in the QRT family, it can in principle only be applied to mappings that belong to that family. However, as we show in this paper, it is still possible to obtain the results of the restoration even when the initial mapping is not of QRT type (at least for the system at hand, but we believe our approach to be of much wider applicability). For the equations derived in this paper we also show how, starting from a form where the independent variable advances one step at a time, one can obtain versions corresponding to multistep evolutions.

math-ph↗

arXiv2018

Detecting discrete integrability: the singularity approach

We describe the various types of singularities that can arise for second order rational mappings and we discuss the historical and present-day, practical, role the singularity confinement property plays as an integrability detector. In particular, we show how singularity analysis can be used to calculate explicitly the dynamical degree for such mappings.

math-ph↗

arXiv2018

Singularity patterns and dynamical degrees

We explain on a selection of mappings how the method introduced by Halburd and our simplified variant thereof, the so-called express method, can be used to calculate the dynamical degree of second-order rational mappings from nothing more than their singularity structure.

math-ph↗

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↗