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Alexander Stokes

Publications and source records attributed to Alexander Stokes.

17 recordsLinked to original sources
arXiv2026

On monodromy of monodromy surfaces

Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlev\'e equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlev\'e equations $\rm{VI},\rm{IV},\rm{II},\rm{I}$. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlev\'e equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlev\'e equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.

math.AG↗
arXiv2026

On the discrete Painlev\'e equivalence problem, non-conjugate translations and nodal curves

We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and Meixner weights. We identify these as discrete Painlev\'e equations and establish their types in the Sakai classification scheme in terms of the associated rational surfaces. In particular, we find examples which come from different weights and share a common surface type $D_5^{(1)}$ but are inequivalent in two ways. First, their dynamics are generated by non-conjugate elements of $\widehat{W}(A_3^{(1)})$. Second, some of the examples have associated surfaces being non-generic in the sense of having nodal curves. The symmetries of these examples form subgroups of the generic symmetry group, which we compute. In particular, we find $(W(A_1^{(1)})\times W(A_1^{(1)}))\rtimes \mathbb{Z}/2\mathbb{Z}$. These examples give further weight to the argument that any correspondence between different weights and the Sakai classification should make use of the refined version of the discrete Painlev\'e equivalence problem, which takes into account not just surface type, but also the group elements generating the dynamics as well as parameter constraints, e.g. those corresponding to nodal curves.

nlin.SI↗
arXiv2025

Classical Algebraic Geometry and Discrete Integrable Systems

The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry side, which is applied to differential and discrete Painlev\'e equations on the integrable systems side. Along the way we also discuss the theory of resolution of indeterminacies, which is applied to the cohomological computation of algebraic entropy of birational transformations of projective spaces, which is closely related to the integrability of the discrete systems they define.

math.AG↗
arXiv2025

On the geometry of a 4-dimensional extension of a $q$-Painlev\'e I equation with symmetry type $A_1^{(1)}$

We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a $q$-Painlev\'e I equation with symmetry of type $A_1^{(1)}$. By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from $({\mathbb P}^1)^{\times 4}$ by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group $\widetilde{W}(A_1^{(1)})\times \widetilde{W}(A_1^{(1)})$ by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of $q$-Painlev\'e equations, one of which is a deautonomisation of the original autonomous integrable map.

nlin.SI↗
arXiv2025

What is the symmetry group of a d-${\text{P}_{\mathrm{II}}}$ discrete Painlev\'e equation?

The symmetry group of a (discrete) Painlev\'e equation provides crucial information on the properties of the equation. In this paper we argue against the commonly-held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception on a specific example of a d-${\text{P}_{\mathrm{II}}}$ equation which corresponds to a half-translation on the root lattice dual to its surface-type root lattice, but which becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways: first by a brute force calculation and second through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized.

nlin.SI↗
arXiv2024

The Painlev\'e equivalence problem for a constrained 3D system

In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential equations arising in the theory of semi-classical orthogonal polynomials. We show that it can be restricted to a system of two first-order differential equations in two different ways on an invariant hypersurface. We build the space of initial conditions for each of these restricted systems and verify that they exhibit the Painlev\'e property from a geometric perspective. Utilising the Painlev\'e identification algorithm we also relate this system to the Painlev\'e VI equation and we build its global Hamiltonian structure. Finally, we prove that the autonomous limit of the original system is Liouville integrable, and the level curves of its first integrals are elliptic curves, which leads us to conjecture that the 3D system itself also possesses the Painlev\'e property without the need to restrict it to the invariant hypersurface.

nlin.SI↗
arXiv2024

Flight-Scope: microscopy with microfluidics in microgravity

With the European Space Agency (ESA) and NASA working to return humans to the moon and onwards to Mars, it has never been more important to study the impact of altered gravity conditions on biological organisms. These include astronauts but also useful micro-organisms they may bring with them to produce food, medicine, and other useful compounds by synthetic biology. Parabolic flights are one of the most accessible microgravity research platforms but present their own challenges: relatively short periods of altered gravity (~20s) and aircraft vibration. Live-imaging is necessary in these altered-gravity conditions to readout any real-time phenotypes. Here we present Flight-Scope, a new microscopy and microfluidics platform to study dynamic cellular processes during the short, altered gravity periods on parabolic flights. We demonstrated Flight-Scopes capability by performing live and dynamic imaging of fluorescent glucose uptake by yeast, S. cerevisiae, on board an ESA parabolic flight. Flight-Scope operated well in this challenging environment, opening the way for future microgravity experiments on biological organisms.

physics.bio-ph↗
arXiv2024

On real and imaginary roots of generalised Okamoto polynomials

Recently, B. Yang and J. Yang derived a family of rational solutions to the Sasa-Satsuma equation, and showed that any of its members constitutes a partial-rogue wave provided that an associated generalised Okamoto polynomial has no real roots or no imaginary roots. In this paper, we derive exact formulas for the number of real and the number of imaginary roots of the generalised Okamoto polynomials. On the one hand, this yields a list of partial-rogue waves that satisfy the Sasa-Satsuma equation. On the other hand, it gives families of rational solutions of the fourth Painlev\'e equation that are pole-free on either the real line or the imaginary line. To obtain these formulas, we develop an algorithmic procedure to derive the qualitative distribution of singularities on the real line for real solutions of Painlev\'e equations, starting from the known distribution for a seed solution, through the action of B\"acklund transformations on the rational surfaces forming their spaces of initial conditions.

nlin.SI↗
arXiv2024

Delay Painlev\'e-I equation, associated polynomials and Masur-Veech volumes

We study a delay-differential analogue of the first Painlev\'e equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlev\'e-I equation in terms of an affine Weyl group of type $A_1^{(1)}$. As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas.

nlin.SI↗
arXiv2023

Deautonomisation by singularity confinement and degree growth

In this paper we give an explanation of a number of observations relating to degree growth of birational mappings of the plane and their deautonomisation by singularity confinement. These observations are of a link between two a priori unrelated notions: firstly the dynamical degree of the mapping and secondly the evolution of parameters required for its singularity structure to remain unchanged under a sufficiently general deautonomisation. We explain this correspondence for a large class of birational mappings of the plane via the spaces of initial conditions for their deautonomised versions. We show that even for non-integrable mappings in this class, the surfaces forming these spaces have effective anticanonical divisors and one can define a period map parametrising them, similar to that in the theory of rational surfaces associated with discrete Painlev\'e equations. This provides a bridge between the evolution of coefficients in the deautonomised mapping and the induced dynamics on the Picard lattice which encode the dynamical degree.

nlin.SI↗
arXiv2022

Takasaki's rational fourth Painlev\'e-Calogero system and geometric regularisability of algebro-Painlev\'e equations

We study a Hamiltonian system without the Painlev\'e property and show that it admits a kind of regularisation on a bundle of rational surfaces with certain divisors removed, generalising Okamoto's spaces of initial conditions for the Painlev\'e differential equations. The system in question was obtained by Takasaki as part of the Painlev\'e-Calogero correspondence and possesses the algebro-Painlev\'e property, being related by an algebraic transformation to the fourth Painlev\'e equation. We provide an atlas for the bundle of surfaces in which the system has a global Hamiltonian structure, with all Hamiltonian functions being polynomial in coordinates just as in the case of Okamoto's spaces. We compare the surface associated with the Takasaki system with that of the fourth Painlev\'e equation, showing that they are related by a combination of blowdowns and a branched double cover, under which we lift the birational B\"acklund transformation symmetries of the fourth Painlev\'e equation to algebraic ones of the Takasaki system, including a discrete Painlev\'e equation. We also discuss and provide more examples in support of the idea that there is a connection between the algebro-Painlev\'e property and similar notions of regularisability, in an analogous way to how regular initial value problems for the Painlev\'e equations everywhere on Okamoto's spaces are related to the Painlev\'e property.

nlin.SI↗
arXiv2021

Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlev\'e equations via the geometric approach

In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlev\'e equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlev\'e equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrence coefficients to the sixth Painlev\'e equation, extending the results of \cite{DFS19}, where a similar approach was used for a discrete system for the same recurrence coefficients. Though the discrete and differential systems here share the same geometry, the construction of the space of initial conditions from the differential system is different and reveals extra considerations that must be made. We also discuss a number of related topics in the context of the geometric approach, such as Hamiltonian forms of the differential equations for the recurrence coefficients, Riccati solutions for special parameter values, and associated discrete Painlev\'e equations.

math.CA↗
arXiv2021

Different Hamiltonians for differential Painlev\'e equations and their identification using a geometric approach

It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same differential Painlev\'e equation. Recognizing a Painlev\'e equation, for example when it appears in some applied problem, is known as the \emph{Painlev\'e equivalence problem}, and the question that we consider here is the Hamiltonian form of this problem. Making such identification explicit, on the level of coordinate transformations, can be very helpful for an applied problem, since it gives access to the wealth of known results about Painlev\'e equations, such as the structure of the symmetry group, special solutions for special values of the parameters, and so on. It can also provide an explicit link between different problems that have the same underlying structure. In this paper we describe a systematic procedure for finding changes of coordinates that trasform different Hamiltonian representations of a Painlev\'e equation into some chosen canonical form. Our approach is based on Sakai's geometric theory of Painlev\'e equations. We explain this procedure in detail for the fourth differential ${\text{P}_{\mathrm{IV}}}$ equation, and also give a brief summary of some known examples for ${\text{P}_{\mathrm{V}}}$ and ${\text{P}_{\mathrm{VI}}}$ cases. It is clear that this approach can easily be adapted to other examples as well, so we expect our paper to be a useful reference for some of the realizations of Okamoto spaces of initial conditions for Painlev\'e equations.

nlin.SI↗
arXiv2020

Singularity confinement in delay-differential Painlev\'e equations

We study singularity confinement phenomena in examples of delay-differential Painlev\'e equations, which involve shifts and derivatives with respect to a single independent variable. We propose a geometric interpretation of our results in terms of mappings between jet spaces, defining certain singularities analogous to those of interest in the singularity analysis of discrete systems, and what it means for them to be confined. For three previously studied examples of delay-differential Painlev\'e equations, we describe all such singularities and show they are confined in the sense of our geometric description.

nlin.SI↗
arXiv2019

Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlev\'e equations

Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answering this question. In this paper we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlev\'e equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for discrete orthogonal polynomials can be expressed in terms of solutions of discrete Painlev\'e equations. In this work we study discrete orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlev\'e-V equation. We also provide an explicit change of variables transforming this equation to the standard form.

nlin.SI↗
arXiv2018

Full-parameter discrete Painlev\'e systems from non-translational Cremona isometries

Since the classification of discrete Painlev\'e equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list. For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries. Translation elements of this symmetry group define discrete Painlev\'e equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlev\'e-type through projective reduction, these have fewer than the maximal number of free parameters corresponding to their surface type. We show that difference equations with the full number of free parameters can be constructed from non-translation elements of infinite order in the symmetry group, constructing several examples and demonstrating their integrability. This is prompted by the study of a previously proposed discrete Painlev\'e equation related to a special class of discrete analogues of surfaces of constant negative Gaussian curvature, which we generalise to a full-parameter integrable difference equation, given by the Cremona action of a non-translation element of the extended affine Weyl group $\widetilde{W}(D_4^{(1)})$ on a family of generic $D_4^{(1)}$- surfaces.

nlin.SI↗