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Discrete equations from Bäcklund transformations of the fifth Painlevé equation

Peter A. Clarkson, Clare Dunning, Ben Mitchell

TL;DR

This work derives discrete Painlevé equations from Bäcklund transformations of the fifth Painlevé equation $P_{ m V}$ and introduces a novel ternary-symmetric discrete equation. It builds explicit hierarchies of rational solutions by leveraging two determinant-based families of polynomials: generalised Laguerre polynomials and generalised Umemura polynomials, both expressible as Wronskians. A central feature is that nonuniqueness of seed rational $P_{ m V}$ solutions yields distinct hierarchies of solutions that satisfy the same discrete equations, enriching the solution structure beyond continuum limits. The results illuminate deep connections between discrete dynamics, orthogonal polynomials, and the affine Weyl-group symmetries underlying $P_{ m V}$, with potential implications for related integrable systems and random matrix contexts.

Abstract

In this paper discrete equations are derived from Bäcklund transformations of the fifth Painlevé equation, including a new discrete equation which has ternary symmetry. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials and the other in terms of the generalised Umemura polynomials, both of which can be expressed as Wronskians of Laguerre polynomials. Hierarchies of rational solutions of the discrete equations are derived in terms of the generalised Laguerre and generalised Umemura polynomials. It is known that there is nonuniqueness of some rational solutions of the fifth Painlevé equation. Pairs of nonunique rational solutions are used to derive distinct hierarchies of rational solutions which satisfy the same discrete equation.

Discrete equations from Bäcklund transformations of the fifth Painlevé equation

TL;DR

This work derives discrete Painlevé equations from Bäcklund transformations of the fifth Painlevé equation and introduces a novel ternary-symmetric discrete equation. It builds explicit hierarchies of rational solutions by leveraging two determinant-based families of polynomials: generalised Laguerre polynomials and generalised Umemura polynomials, both expressible as Wronskians. A central feature is that nonuniqueness of seed rational solutions yields distinct hierarchies of solutions that satisfy the same discrete equations, enriching the solution structure beyond continuum limits. The results illuminate deep connections between discrete dynamics, orthogonal polynomials, and the affine Weyl-group symmetries underlying , with potential implications for related integrable systems and random matrix contexts.

Abstract

In this paper discrete equations are derived from Bäcklund transformations of the fifth Painlevé equation, including a new discrete equation which has ternary symmetry. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials and the other in terms of the generalised Umemura polynomials, both of which can be expressed as Wronskians of Laguerre polynomials. Hierarchies of rational solutions of the discrete equations are derived in terms of the generalised Laguerre and generalised Umemura polynomials. It is known that there is nonuniqueness of some rational solutions of the fifth Painlevé equation. Pairs of nonunique rational solutions are used to derive distinct hierarchies of rational solutions which satisfy the same discrete equation.
Paper Structure (30 sections, 28 theorems, 337 equations, 3 tables)

This paper contains 30 sections, 28 theorems, 337 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Bäck-lund transformations of the fifth Painlevé equation
  3. Deriving discrete Painlevé equations
  4. First discrete equation: asymmetric discrete Painlevé II
  5. Second discrete equation
  6. Third discrete equation: ternary discrete P$_{\rm I}$
  7. Fourth discrete equation: new equation with ternary symmetry
  8. Differential-difference equations
  9. Rational solutions of P$_{\rm V}$
  10. Generalised Laguerre polynomials
  11. Generalised Umemura polynomials
  12. Rational solutions of the discrete equations
  13. Rational solutions in terms of generalised Laguerre polynomials
  14. First discrete equation: asymmetric discrete P$_{\rm II}$
  15. Second discrete equation
  16. ...and 15 more sections

Key Result

Lemma 3.4

Suppose that $x_{n}$, $x_{n+2}$ and $x_{n-2}$ are solutions of the ternary dP$_{\rm I}$ equation eq:tdPI, then they satisfy the discrete equation where $a_{n}$ is given by eq:abcn3a, i.e. with $\lambda$, $\rho$ and $\varphi$ arbitrary constants.

Theorems & Definitions (76)

  • Definition 2.1
  • Definition 2.2
  • Remark 3.2
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • proof
  • Remark 3.6
  • Theorem 4.1
  • proof
  • ...and 66 more