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Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

John Gibbons, Alexander Stokes, Alexander P. Veselov

Abstract

We study a delay-differential analogue of the first Painlevé 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 Painlevé-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.

Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

Abstract

We study a delay-differential analogue of the first Painlevé 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 Painlevé-I equation in terms of an affine Weyl group of type . 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.
Paper Structure (13 sections, 17 theorems, 165 equations, 6 figures)

This paper contains 13 sections, 17 theorems, 165 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The conservation form of the equation
  3. Formal entire solutions
  4. Polynomials from the trivial seed
  5. Conserved quantity and parameters $\tau$
  6. Polynomials $\Phi_n(z)$ with all $\tau_i=0$
  7. Outer expansion
  8. Special case $s_i=0$: Bernoulli-Catalan polynomials
  9. Meromorphic solutions and singularity structure
  10. Singularity confinement
  11. An affine Weyl group structure of singularity patterns
  12. Link with the calculation of Masur-Veech volumes
  13. Concluding remarks

Key Result

Proposition 2.1

The delay Painlevé-I equation delayP1 can be rewritten in the conservation form where In other words, $S$ is a conserved quantity of the delay Painlevé-I equation.

Figures (6)

  • Figure 1: Plots of $\Phi_n(z)$ for real $z$
  • Figure 2: Plots of Bernoulli polynomials $B_n(x)$ for real $x$
  • Figure 3: Plots of Euler polynomials $E_n(x)$ for real $x$
  • Figure 4: Zeroes of $\Phi_n(z)$
  • Figure 5: Zeroes of $B_n(x)$
  • ...and 1 more figures

Theorems & Definitions (25)

  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • ...and 15 more