Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

J. W. E. Harrow, A. N. W. Hone

Abstract

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 $D_7$.

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

Abstract

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 .

Paper Structure

This paper contains 7 sections, 13 theorems, 250 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Confluent Darboux transformations
  3. Yablonskii-Vorob'ev polynomials via Adler and Moser
  4. Ohyama polynomials and BTs for Painlevé III ($D_7$)
  5. Proof of main theorem
  6. Lax pair for Painlevé III ($D7$) and generating functions
  7. Conclusions

Key Result

Theorem 1.1

For the values $\ell=n+1/2$ of the parameter in the Hamiltonian (piihell), the Painlevé II equation (pii) has rational solutions given by where the tau function associated with $h_{n+1/2}$ is given by for integer $n\geq 1$, with $y_{n+1/2}=-y_{-n-1/2}$ for $n\leq 0$, and polynomials $p_k(z)$ defined by the generating function

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Proposition 4.3
  • proof
  • Remark 4.4
  • ...and 16 more