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Different Hamiltonians for differential Painlevé equations and their identification using a geometric approach

Anton Dzhamay, Galina Filipuk, Adam Ligȩza, Alexander Stokes

Abstract

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same differential Painlevé equation. Recognizing a Painlevé equation, for example when it appears in some applied problem, is known as the \emph{Painlevé equivalence problem}, and the question that we consider here is the Hamiltonian form of this problem. Making such identification explicit, on the level of coordinate transformations, can be very helpful for an applied problem, since it gives access to the wealth of known results about Painlevé equations, such as the structure of the symmetry group, special solutions for special values of the parameters, and so on. It can also provide an explicit link between different problems that have the same underlying structure. In this paper we describe a systematic procedure for finding changes of coordinates that trasform different Hamiltonian representations of a Painlevé equation into some chosen canonical form. Our approach is based on Sakai's geometric theory of Painlevé equations. We explain this procedure in detail for the fourth differential ${\text{P}_{\mathrm{IV}}}$ equation, and also give a brief summary of some known examples for ${\text{P}_{\mathrm{V}}}$ and ${\text{P}_{\mathrm{VI}}}$ cases. It is clear that this approach can easily be adapted to other examples as well, so we expect our paper to be a useful reference for some of the realizations of Okamoto spaces of initial conditions for Painlevé equations.

Different Hamiltonians for differential Painlevé equations and their identification using a geometric approach

Abstract

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same differential Painlevé equation. Recognizing a Painlevé equation, for example when it appears in some applied problem, is known as the \emph{Painlevé equivalence problem}, and the question that we consider here is the Hamiltonian form of this problem. Making such identification explicit, on the level of coordinate transformations, can be very helpful for an applied problem, since it gives access to the wealth of known results about Painlevé equations, such as the structure of the symmetry group, special solutions for special values of the parameters, and so on. It can also provide an explicit link between different problems that have the same underlying structure. In this paper we describe a systematic procedure for finding changes of coordinates that trasform different Hamiltonian representations of a Painlevé equation into some chosen canonical form. Our approach is based on Sakai's geometric theory of Painlevé equations. We explain this procedure in detail for the fourth differential equation, and also give a brief summary of some known examples for and cases. It is clear that this approach can easily be adapted to other examples as well, so we expect our paper to be a useful reference for some of the realizations of Okamoto spaces of initial conditions for Painlevé equations.

Paper Structure

This paper contains 32 sections, 22 theorems, 174 equations, 20 figures.

Table of Contents

  1. Introduction
  2. The Fourth Painevé Equation $\text{P}_{\mathrm{IV}}$ and its Hamiltonian Forms
  3. Its--Prokhorov and Jimbo-Miwa Hamiltonians
  4. Okamoto Hamiltonian
  5. Filipuk-Żoła̧dek Hamiltonian
  6. Kecker Hamiltonian
  7. Coordinate transformations between different Hamiltonian forms for $\text{P}_{\mathrm{IV}}$
  8. Its-Prokhorov and Jimbo-Miwa systems
  9. The Kecker system
  10. Preliminaries
  11. The Okamoto space of initial conditions
  12. Geometry of the standard $E_{6}^{(1)}$ Surface
  13. The point configuration
  14. The period map and the root variables
  15. The extended affine Weyl symmetry group
  16. ...and 17 more sections

Key Result

Theorem 1

The Its-Prokhorov Hamiltonian system eq:IP-Ham-4 and the Okamoto Hamiltonian system eq:Ok-Ham-4 are related by the following change of variables and parameter correspondence: The Hamiltonians are then related by where the purely $t$-dependent terms can in fact be ignored.

Figures (20)

  • Figure 1: Relationship between different Hamiltonian Systems for $\text{P}_{\mathrm{IV}}$
  • Figure 2: The Blowup Procedure
  • Figure 3: The Space of Initial Conditions for the $\text{P}_{\mathrm{IV}}$ Okamoto Hamiltonian System \ref{['eq:Ok-Ham-4']}
  • Figure 4: The Surface Root Basis for the standard $E_{6}^{(1)}$ Sakai surface
  • Figure 5: The standard $E_{6}^{(1)}$ Sakai surface
  • ...and 15 more figures

Theorems & Definitions (35)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 3
  • Remark 4
  • ...and 25 more