Variations for Some Painlevé Equations
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Fecha
2019
Autores
Acosta-Humañez, Primitivo B.
Van Der Put, Marius
Top, Jaap
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Editor
SIGMA
Resumen
This paper rst discusses irreducibility of a Painlev e equation P. We explain
how the Painlev e property is helpful for the computation of special classical and algebraic
solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian H to
a Painlev e equation P. Complete integrability of H is shown to imply that all solutions
to P are classical (which includes algebraic), so in particular P is solvable by \quadratures".
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 P2 to P5 where algebraic solutions are present, by
showing how our results lead to a quick computation of the component of the identity of
the di erential Galois group for the rst two variational equations. As expected there are
no cases where this group is commutative.
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Palabras clave
Hamiltonian systems, Variational equations, Painlevé equations, differential Galois groups