On the Convergence of Formal Exotic Series Solutions of an ODE

Gontsov, Renat; Goryuchkina, Irina

doi:10.1007/s40315-020-00306-z

Cited by 1 publication

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“…The latter have also been studied in [44,45,46], where convergence has been proved. See also [2,3,4] for expansions of PVI transendents, and [17,18,19,20,21,57] for convergence issues.…”

Section: Introductionmentioning

confidence: 99%

The sixth Painleve' equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

Gabriele¹,

Guzzetti²

2021

Preprint

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We consider a 3-dimensional Pfaffian system, whose z-component is a differential system with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlevé equation PVI. The coefficients of the system will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [48] and correct some misprints there. Then, we express in terms of the Stokes matrices of the irregular system the monodromy invariants p jk " TrpM j M k q of a 2-dimensional Fuchsian system with four singularities, traditionally employed in the isomonodromic deformation method for PVI. The representation of PVI as integrability condition for the above mentioned 3-dimensional Pfaffian system naturally allows to classify the branches of PVI transcendents which are holomorphic at a critical point, and such that the Pfaffian system satisfies the analyticity and semisimplicity properties described in [7]. These properties make the explicit computation possible of the monodromy data associated to the classified transcendents: we compute both the associated Stokes matrices and the invariants p jk . Finally, we compute the monodromy data parametrizing the chamber of a Dubrovin-Frobenius manifold associated with a transcendent holomorphic at x " 0.

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“…The latter have also been studied in [44,45,46], where convergence has been proved. See also [2,3,4] for expansions of PVI transendents, and [17,18,19,20,21,57] for convergence issues.…”

Section: Introductionmentioning

confidence: 99%