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Unified structures for solutions of Painlevé equation II and Somos-4 like relations for the tau functions

Federico Zullo, Maria Grazia Naso, Elena Vuk

TL;DR

This work develops a unified tau-function framework that ties together Painleve II, its Hamiltonian structure, and the related Painleve XXXIV dynamics. By exploiting Bäcklund transformations, it shows that Painleve II, its Hamiltonians, and the tau functions can be expressed as rational functions of tau functions, with Wronskian and derivative relations following suit, and that the tau functions satisfy a non-autonomous Somos-4 type relation. The authors explore degenerate limits including Weierstrass elliptic, Yablonskii-Vorob'ev polynomials, and Airy-type solutions, providing explicit tau-function realizations in each case and deriving multiple linear, bilinear, and recursive relations that illuminate the algebraic backbone of the hierarchy. The results offer a cohesive algebraic perspective on the Painleve II family, suggesting avenues for extension to higher Painleve equations and connections to determinant representations and broader integrable systems.

Abstract

We present certain general structures related to the solutions of Painlevé equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking advantage of the Bäcklund transformations we find different explicit rational expressions linking the solutions of Painlevé equation II, Painlevé equation XXXIV and the Hamiltonians with the tau functions. Wronskians among different tau functions and the derivatives of the tau functions themselves will be expressed in terms of rational functions of tau functions too. A non-autonomous Somos-4 type relation solved by these functions is given. For the Somos-4 type relation we consider degenerate cases through the use of suitable parameters inserted into the equations: the autonomous case solvable in terms of Weierstrass elliptic functions, the case corresponding to the Yablonskii-Vorob'ev polynomials, the Airy-type solutions and the more general transcendental case.

Unified structures for solutions of Painlevé equation II and Somos-4 like relations for the tau functions

TL;DR

This work develops a unified tau-function framework that ties together Painleve II, its Hamiltonian structure, and the related Painleve XXXIV dynamics. By exploiting Bäcklund transformations, it shows that Painleve II, its Hamiltonians, and the tau functions can be expressed as rational functions of tau functions, with Wronskian and derivative relations following suit, and that the tau functions satisfy a non-autonomous Somos-4 type relation. The authors explore degenerate limits including Weierstrass elliptic, Yablonskii-Vorob'ev polynomials, and Airy-type solutions, providing explicit tau-function realizations in each case and deriving multiple linear, bilinear, and recursive relations that illuminate the algebraic backbone of the hierarchy. The results offer a cohesive algebraic perspective on the Painleve II family, suggesting avenues for extension to higher Painleve equations and connections to determinant representations and broader integrable systems.

Abstract

We present certain general structures related to the solutions of Painlevé equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking advantage of the Bäcklund transformations we find different explicit rational expressions linking the solutions of Painlevé equation II, Painlevé equation XXXIV and the Hamiltonians with the tau functions. Wronskians among different tau functions and the derivatives of the tau functions themselves will be expressed in terms of rational functions of tau functions too. A non-autonomous Somos-4 type relation solved by these functions is given. For the Somos-4 type relation we consider degenerate cases through the use of suitable parameters inserted into the equations: the autonomous case solvable in terms of Weierstrass elliptic functions, the case corresponding to the Yablonskii-Vorob'ev polynomials, the Airy-type solutions and the more general transcendental case.
Paper Structure (12 sections, 9 theorems, 137 equations, 1 figure)

This paper contains 12 sections, 9 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Canonical transformations
  3. Recursions for the Hamiltonian functions
  4. Linearization of the solutions of the Painlevé II equation
  5. The ubiquitous role of the tau functions
  6. A Somos-4 like relation for the $\tau$ functions
  7. The Weierstrass case
  8. The case of Yablonskii-Vorob'ev polynomials
  9. The Airy-type solutions
  10. The general case
  11. Wronskian relations
  12. Conclusions

Key Result

Proposition 6.2

The function $\tau_n$ divides $\alpha_n^2\tau_{n+1}^3+\beta_{n-1}\tau_{n-1}\tau_{n+2}^2$, whereas the function $\tau_{n+2}$ divides $\alpha_n^2\tau_{n+1}^3+\beta_{n+1}\tau_n^2\tau_{n+3}$. More precisely one has

Figures (1)

  • Figure 1: On the left we plot of the real and imaginary part of the roots of $Q_5$ (red) and $P_5$ (black), on the right the roots of $Q_{17}$ (red) and $P_{17}$ (black).

Theorems & Definitions (11)

  • Definition 6.1
  • Proposition 6.2
  • Proposition 6.3
  • Proposition 6.4
  • Proposition 6.5
  • Remark 6.6
  • Corollary 6.7
  • Proposition 6.8
  • Corollary 6.9
  • Proposition 6.10
  • ...and 1 more