arXiv subjects

A. N. W. Hone

Publications and source records attributed to A. N. W. Hone.

At least 19 recordsLinked to original sources

arXiv2026

Commuting integrable maps from a deformed D$_4$ cluster algebra

In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D$_4$. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the $q$-Painlevé VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up $\mathbb{P}^1 \times \mathbb{P}^1$.

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arXiv2025

Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlevé III ($D_7$) equation

We describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii-Vorob'ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$.

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arXiv2024

Casting more light in the shadows: dual Somos-5 sequences

Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of dual numbers. Furthermore, by iterating recurrences with specific numerical values, some particular well-known integer sequences, such as the Fibonacci sequence, Markoff numbers, and Somos sequences, were shown to produce associated ``shadow'' sequences when they were extended to the dual numbers. Here we consider the most general version of the Somos-5 recurrence defined over the ring of dual numbers $\mathbb{D}$ with complex coefficients, that is the ring $\mathbb{C}[\varepsilon]$ modulo the relation $\varepsilon^2=0$. We present three different ways to present the general solution of the initial value problem for Somos-5 and its shadow part: in analytic form, using the Weierstrass sigma function with arguments in $\mathbb{D}$; in terms of the solution of a linear difference equation; and using Hankel determinants constructed from $\mathbb{D}$-valued moments, via a connection with a Quispel-Roberts-Thompson (QRT) map over the dual numbers.

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arXiv2024

A family of integrable maps associated with the Volterra lattice

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the $g=2$ case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g\geqslant 1$. The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case ($g=1$), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices.

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arXiv2024

Integrable maps in 4D and modified Volterra lattices

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus $g$ algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted ${\cal V}_g$. The original motivation behind our work was the fact that, in the particular case $g=2$, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map ${\cal V}_2$ yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of the other 4D integrable maps obtained in the classification of Gubbiotti et al. correspond to genus two solutions of two different forms of the modified Volterra lattice, being related via a Miura-type transformation to the $g=2$ Volterra map ${\cal V}_2$. We dedicate this work to a dear friend and colleague, Decio Levi.

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arXiv2022

Similarity reductions of peakon equations: integrable cubic equations

We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa-Holm (mCH) equation and Novikov's equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov's equation to a negative mKdV flow and a negative Sawada-Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlevé type: for the mCH equation, its reduction is of second order and second degree, while for Novikov's equation the reduction is a particular case of Painlevé V. Furthermore, we show that each of these two different Painlevé-type equations is related to the particular cases of Painlevé III that arise from analogous similarity reductions of the Camassa-Holm and the Degasperis-Procesi equation, respectively. For each of the cubically nonlinear PDEs considered, we also give explicit parametric forms of their periodic travelling wave solutions in terms of elliptic functions. We present some parametric plots of the latter, and, by using explicit algebraic solutions of Painlevé III, we do the same for some of the simplest examples of scaling similarity solutions, together with descriptions of their leading order asymptotic behaviour.

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arXiv2019

Analogues of Kahan's method for higher order equations of higher degree

Kahan introduced an explicit method of discretization for systems of first order differential equations with nonlinearities of degree at most two (quadratic vector fields). Kahan's method has attracted much interest due to the fact that it preserves many of the geometrical properties of the original continuous system. In particular, a large number of Hamiltonian systems of quadratic vector fields are known for which their Kahan discretization is a discrete integrable system. In this note, we introduce a special class of explicit order-preserving discretization schemes that are appropriate for certain systems of ordinary differential equations of higher order and higher degree.

math.NA↗

arXiv2019

A new class of integrable Lotka-Volterra systems

A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.

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arXiv2019

Hirota bilinear equations for Painlevé transcendents

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

math.CA↗

arXiv2018

QRT maps and related Laurent systems

In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a fifth-order recurrence with the Laurent property. Here we recursively factorise the 12-parameter symmetric QRT map, given by a second-order recurrence, to obtain a system of three coupled recurrences which possesses the Laurent property. As degenerate special cases, we first derive systems of two coupled recurrences corresponding to the 5-parameter multiplicative and additive symmetric QRT maps. In all cases, the Laurent property is established using a generalisation of a result due to Hickerson, and exact formulae for degree growth are found from ultradiscrete (tropical) analogues of the recurrences. For the general 18-parameter QRT map it is shown that the components of the iterates can be written as a ratio of quantities that satisfy the same Somos-7 recurrence.

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arXiv2017

Some integrable maps and their Hirota bilinear forms

We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

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arXiv2015

Algebraic entropy for algebraic maps

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations.

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arXiv2014

Generic hyperelliptic Prym varieties in a generalized Henon-Heiles system

It is known that the Jacobian of an algebraic curve which is a 2-fold covering of a hyperelliptic curve ramified at two points contains a hyperelliptic Prym variety. Its explicit algebraic description is applied to some of the integrable Henon-Heiles systems with a non-polynomial potential. Namely, we identify the generic complex invariant manifolds of the systems as a hyperelliptic Prym subvariety of the Jacobian of the spectral curve of the corresponding Lax representation. The exact discretization of the system is described as a translation on the Prym variety.

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arXiv2013

A family of linearisable recurrences with the Laurent property

We consider a family of nonlinear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearisable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrodinger operators is also explained.

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arXiv2013

Properties of the series solution for Painleve I

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painleve equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented.

math.CA↗

arXiv2013

Integrability of reductions of the discrete KdV and potential KdV equations

We study the integrability of mappings obtained as reductions of the discrete Korteweg-de Vries (KdV) equation and of two copies of the discrete potential Korteweg-de Vries equation (pKdV). We show that the mappings corresponding to the discrete KdV equation, which can be derived from the latter, are completely integrable in the Liouville-Arnold sense. The mappings associated with two copies of the pKdV equation are also shown to be integrable.

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arXiv2008

Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras

Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system. The Euler top is naturally written in terms of the $\mathfrak{so}(3)$ Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.

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arXiv2008

Algebraic curves, integer sequences and a discrete Painleve transcendent

We consider some bilinear recurrences that have applications in number theory. The explicit solution of a general three-term bilinear recurrence relation of fourth order is given in terms of the Weierstrass sigma function for an associated elliptic curve. The recurrences can generate integer sequences, including the Somos 4 sequence and elliptic divisibility sequences. An interpretation via the theory of integrable systems suggests the relation between certain higher order recurrences and hyperelliptic curves of higher genus. Analogous sequences associated with a $q$-discrete Painlevé I equation are briefly considered.

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