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Nalini Joshi

Publications and source records attributed to Nalini Joshi.

At least 19 recordsLinked to original sources
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

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 Painlev\'e equation and the difference second Painlev\'e 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↗
arXiv2025

Arithmetic dynamics of a discrete Painlev\'e equation

We consider the orbits of a discrete Painlev\'e 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 Painlev\'e equations over $\mathbb{C}$, whose generic solutions are believed to be more transcendental than elliptic functions.

nlin.SI↗
arXiv2024

QRT Map on a Bielliptic Surface

The family of mappings of the plane possessing a biquadratic invariant, which is known collectively as QRT maps, is composed of two involutions, one preserving a vertical shift and the other preserving a horizontal shift in the plane. In this paper, we extend the map by replacing each shift by the group operation on each of two families of elliptic curves, whose product forms a bielliptic surface.

nlin.SI↗
arXiv2024

On the crystal limit of the q-difference sixth Painlev\'e equation

We consider the Riemann-Hilbert correspondence associated with the $q$-difference sixth Painlev\'e 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

Segre surfaces and geometry of the Painlev\'e 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 Painlev\'e 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 Painlev\'e differential equation.

math-ph↗
arXiv2024

Exponentially-improved asymptotics for $q$-difference equations: ${}_2ϕ_0$ and $q{\rm P}_{\rm I}$

Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size $q^{-\frac12 n(n-1)}$, in which $q\in(0,1)$ is fixed. Hence, the divergence is much stronger, and one has to introduce alternative Borel and Laplace transforms to make sense of these formal series. We will discuss exponentially-improved asymptotics for the basic hypergeometric function ${}_2ϕ_0$ and for solutions of the $q$-difference first Painlevé equation $q{\rm P}_{\rm I}$. These are optimal truncated expansions, and re-expansions in terms of new $q$-hyperterminant functions. The re-expansions do incorporate the Stokes phenomena.

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

Asymptotic behaviours of q-orthogonal polynomials from a q-Riemann Hilbert Problem

We describe a Riemann-Hilbert problem for a family of $q$-orthogonal polynomials, $\{ P_n(x) \}_{n=0}^\infty$, and use it to deduce their asymptotic behaviours in the limit as the degree, $n$, approaches infinity. We find that the $q$-orthogonal polynomials studied in this paper share certain universal behaviours in the limit $n\to\infty$. In particular, we observe that the asymptotic behaviour near the location of their smallest zeros, $x \sim q^{n/2}$, and norm, $\|P_n\|_2$, are independent of the weight function as $n\to\infty$.

math.CA↗
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↗
arXiv2022

Asymptotics of Discrete $q$-Freud $\mathrm{II}$ orthogonal polynomials from the $q$-Riemann Hilbert Problem

We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the polynomials approach infinity. The RHP formulation also enables us to obtain further properties. In particular, we consider how the class of polynomials and their asymptotic behaviours change under translations of the $q$-discrete lattice and determine the asymptotics of related $q$-Painlevé equations.

math.CA↗
arXiv2022

Global asymptotics of the sixth Painlevé equation in Okamoto's space

We study dynamics of solutions in the initial value space of the sixth Painlevé equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded, and that the complex limit set of each solution exists and is compact and connected.

nlin.SI↗
arXiv2022

On the Perturbed Second Painlevé Equation

We consider a perturbed version of the second Painlevé equation ($\textrm{P}_{\textrm{II}}$), which arises in applications, and show that it possesses solutions analogous to the celebrated Hastings-McLeod and tritronquée solutions of $\textrm{P}_{\textrm{II}}$. The Hastings-McLeod-type solution of the perturbed equation is holomorphic, real-valued and positive on the whole real-line, while the tritronquée-type solution is holomorphic in a large sector of the complex plane. These properties also characterise the corresponding solutions of $\textrm{P}_{\textrm{II}}$ and are surprising because the perturbed equation does not possess additional distinctive properties that characterise $\textrm{P}_{\textrm{II}}$, particularly the Painlevé property.

math-ph↗
arXiv2021

On a class of q-orthogonal polynomials and the q-Riemann Hilbert Problem

We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the limit as the degree of the polynomial approaches infinity.

math.CA↗
arXiv2021

On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries Equation

Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices.

nlin.SI↗
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↗
arXiv2020

Classification of quad-equations on a cuboctahedron

In this paper, we consider polynomials associated with faces and internal quadrilaterals of a cuboctahedron and classify them under the requirement that they are consistent. These polynomials give rise to a system of partial difference equations on a face-centred cubic lattice. Our results were motivated by $τ$-functions related to discrete Painlevé equations.

nlin.SI↗