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Hamiltonians associated with the sixth Painlevé equation

Abstract

We obtain the formula determining the general form of polynomial Hamiltonians associated with the sixth Painlevé equation and prove its uniqueness. We prove the existence of nonpolynomial Hamiltonians associated with this equation. We identify the Hamiltonian class for which the defining differential equation coincides with the equation (h-equation) for the simplest polynomial Hamiltonian (the Okamoto Hamiltonian).

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 54–65, April, 2007.

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Tsegel’nik, V.V. Hamiltonians associated with the sixth Painlevé equation. Theor Math Phys 151, 482–491 (2007). https://doi.org/10.1007/s11232-007-0036-x

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Keywords

  • Painlevé equation
  • Hamiltonian
  • family of solutions
  • Bäcklund transformation
  • Heun equation