Isomonodromy for the Degenerate Fifth Painlevé Equation

Acosta-Humánez, Primitivo B.; Put, Marius van der; Top, Jaap

doi:10.3842/sigma.2017.029

Cited by 2 publications

(2 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If the line bundle Oe 1 is invariant under ∇, then the same computation provides a contradiction. A similar computation proves the last statement of part (1).…”

Section: Comparison With Work Of Ph Boalchsupporting

confidence: 78%

“…We note that this choice does not do justice to the extensive literature related to Painlevé equations. The family of matrix differential operators for P 3 , P 4 and for the degenerate fifth Painlevé equation degP 5 has been refined, see [1,24,25], to fine moduli spaces of connections on the projective line, which are identified with Okamoto-Painlevé spaces. The detailed construction of the moduli spaces supplements and continues some sections of [22].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Moduli Spaces for the Fifth Painlevé Equation

van der Put,

Top

2023

SIGMA

View full text Add to dashboard Cite

Isomonodromy for the fifth Painlevé equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${\rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${\rm P}_5$, equivalent to the one of Okamoto.

show abstract

“…If the line bundle Oe 1 is invariant under ∇, then the same computation provides a contradiction. A similar computation proves the last statement of part (1).…”

Section: Comparison With Work Of Ph Boalchsupporting

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Moduli Spaces for the Fifth Painlevé Equation

van der Put,

Top

2023

SIGMA

View full text Add to dashboard Cite

show abstract

Variations for Some Painlevé Equations

Acosta-Humánez¹,

Put

Top

2019

SIGMA

Self Cite

View full text Add to dashboard Cite

This paper first discusses irreducibility of a Painleve equation P . We show that reducibility is equivalent to the existence of special classical solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian H to a Painlevé equation P . Complete integrability of H is shown to imply that all solutions to P are special classical or algebraic, so in particular P is reducible.Next, we show that the variational equation of P at a given algebraic solution coincides with the normal variational equation of H at the corresponding solution.Finally, we test the Morales-Ramis theorem in all cases P 2 to P 5 where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the differential Galois group for the first two variational equations. As expected there are no cases where this group is commutative.

show abstract

Isomonodromy for the Degenerate Fifth Painlevé Equation

Cited by 2 publications

References 21 publications

Moduli Spaces for the Fifth Painlevé Equation

Moduli Spaces for the Fifth Painlevé Equation

Variations for Some Painlevé Equations

Contact Info

Product

Resources

About