arXiv subjects

Pieter Roffelsen

Publications and source records attributed to Pieter Roffelsen.

18 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

A Tau function for $q$-Painlev\'e VI as a Fredholm determinant

We give an analytic construction of a tau function for the $q$-difference sixth Painlev\'e equation ($q$PVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding $q$PVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of $q$PVI. Finally, we derive an asymptotic expansion of the tau function for small times $t$.

math-ph↗

arXiv2026

On difference-differential Lax pairs and integrals of Painlev\'e equations in finite characteristic

We collect rank two difference-differential Lax pairs for classical Painlev\'e equations in the literature and put each in $2\times 2$ matrix form with the coefficient matrix of the spectral equation a degree two matrix polynomial. We describe and apply a general method to obtain integrals of motion in characteristic $p$ from these Lax pairs. For every relevant Painlev\'e equation, this leads to a countable list of integrals of motion, with one entry for each prime $p$.

nlin.SI↗

arXiv2026

Segre surfaces and geometry of the Painlevé equations

In this paper, we consider a six parameter family of affine Segre surfaces embedded in $\mathbb C^6$. For generic values of the parameters, this family is associated to the $q$-difference sixth Painlevé equation. We show that different limiting forms of this family give Segre surfaces that are isomorphic as affine varieties to the the monodromy manifolds of each Painlevé differential equation.

math-ph↗

arXiv2026

On integrals of non-autonomous dynamical systems in finite characteristic

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.

nlin.SI↗

arXiv2026

Arithmetic dynamics of a discrete Painlevé equation

We consider the orbits of a discrete Painlevé equation over finite fields and show that the number of points in such orbits satisfy the Hasse bound. The orbits turn out to lie on algebraic curves, whose defining polynomials are given explicitly. Moreover, these curves are shown to have genus less than or equal to one, which contrasts sharply with the case of discrete Painlevé equations over $\mathbb{C}$, whose generic solutions are believed to be more transcendental than elliptic functions.

nlin.SI↗

arXiv2025

On the crystal limit of the q-difference sixth Painlevé equation

We consider the Riemann-Hilbert correspondence associated with the $q$-difference sixth Painlevé equation in the crystal limit, i.e. $q\rightarrow 0$, and show two main results. First, the limit of this generically highly transcendental mapping is shown to exist. Second, we show that the limiting map is bi-rational and describe it explicitly.

nlin.SI↗

arXiv2024

On $q$-Painlevé VI and the geometry of Segre surfaces

In the context of $q$-Painlevé VI with generic parameter values, the Riemann-Hilbert correspondence induces a one-to-one mapping between solutions of the nonlinear equation and points on an affine Segre surface. Upon fixing a generic point on the surface, we give formulae for the function values of the corresponding solution near the critical points, in the form of complete, convergent, asymptotic expansions. These lead in particular to the solution of the nonlinear connection problem for the general solution of $q$-Painlevé VI. We further show that, when the point on the Segre surface is moved to one of the sixteen lines on the surface, one of the asymptotic expansions near the critical points truncates, under suitable parameter assumptions. At intersection points of lines, this then yields doubly truncated asymptotics at one of the critical points or simultaneous truncation at both.

nlin.SI↗

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 Painlevé 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 Painlevé equations, starting from the known distribution for a seed solution, through the action of Bäcklund transformations on the rational surfaces forming their spaces of initial conditions.

nlin.SI↗

arXiv2023

On a class of elliptic orthogonal polynomials and their integrability

Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann-Hilbert framework. We then focus on the special case, defined by a constant weight function, and use the Riemann-Hilbert problem to derive recurrence relations and differential equations for the orthogonal polynomials. We further show that the sub-class of even polynomials is associated to the elliptic form of Painlevé VI, with the tau function given by the Hankel determinant of even moments, up to a scaling factor. The first iteration of these even polynomials relates to the special case of Painlevé VI studied by Hitchin in relation to self-dual Einstein metrics.

math.CA↗

arXiv2023

On symmetric solutions of the fourth $q$-Painlevé equation

The Painlevé equations possess transcendental solutions $y(t)$ with special initial values that are symmetric under rotation or reflection in the complex $t$-plane. They correspond to monodromy problems that are explicitly solvable in terms of classical special functions. In this paper, we show the existence of such solutions for a $q$-difference Painlevé equation. We focus on symmetric solutions of a $q$-difference equation known as $q\textrm{P}_{\textrm{IV}}$ or $q{\rm P}(A_5^{(1)})$ and provide their symmetry properties and solve the corresponding monodromy problem.

nlin.SI↗

arXiv2023

On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem

We study the sixth $q$-difference Painlevé equation ($q{\textrm{P}_{\textrm{VI}}}$) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of $q{\textrm{P}_{\textrm{VI}}}$ with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explicitly in terms of certain orthogonal polynomials yielding special function solutions of $q{\textrm{P}_{\textrm{VI}}}$.

math-ph↗

arXiv2021

Roots of generalised Hermite polynomials when both parameters are large

We study the roots of the generalised Hermite polynomials $H_{m,n}$ when both $m$ and $n$ are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular lattice, as was numerically observed by Clarkson. We describe the elliptic region and the deformed lattice in terms of elliptic integrals and their degenerations. Keywords: Generalised Hermite polynomials; roots asymptotics; Painleve IV; Boutroux Curves; Tritronquee solution.

math.CA↗

arXiv2021

On the Riemann-Hilbert problem for a $q$-difference Painlevé equation

A Riemann-Hilbert problem for a $q$-difference Painlevé equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q\textrm{P}_{\textrm{IV}}$ explicitly.

nlin.SI↗

arXiv2018

Poles of Painlevé IV Rationals and their Distribution

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when $m$ is large and $n$ fixed.

math.CA↗

arXiv2016

Analytic solutions of $q$-$P(A_1)$ near its critical points

For transcendental functions that solve non-linear $q$-difference equations, the best descriptions available are the ones obtained by expansion near critical points at the origin and infinity. We describe such solutions of a $q$-discrete Painlevé equation, with 7 parameters whose initial value space is a rational surface of type $A_1^{(1)}$. The resultant expansions are shown to approach series expansions of the classical sixth Painlevé equation in the continuum limit.

nlin.SI↗

arXiv2012

On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials

We study the real roots of the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlevé equation. It has been conjectured that the number of real roots of the nth Yablonskii-Vorob'ev polynomial equals [(n+1)/2]. We prove this conjecture using an interlacing property between the roots of the Yablonskii-Vorob'ev polynomials. Furthermore we determine precisely the number of negative and the number of positive real roots of the nth Yablonskii-Vorob'ev polynomial.

math.CA↗

arXiv2010

Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them

We study the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlevé equation. Divisibility properties of the coefficients of these polynomials, concerning powers of 4, are obtained and we prove that the nonzero roots of the Yablonskii-Vorob'ev polynomials are irrational. Furthermore, relations between the roots of these polynomials for consecutive degree are found by considering power series expansions of rational solutions of the second Painlevé equation.

math.CA↗