Presented By: Integrable Systems and Random Matrix Theory Seminar - Department of Mathematics
ISRMT seminar: On the exact Gevrey order of formal Puiseux series solutions to the third Painlevé equation
Anastasia Parusnikova (HSE University)
I would like to speak about one rather old joint work with Andrey Vasilyev.
We consider the third Painlevé equation.
The Puiseux series formally satisfying it, asymptotically approximate of Gevrey order one solutions to this equation in sectors with the vertices at infinity.
A condition sufficient for the convergence of formal solutions of an ODE with analytic left-hand side is well-known.
On the other hand sufficient conditions for the divergence given in the same terms are unknown.
We present the family of values of the parameters such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.
We consider the third Painlevé equation.
The Puiseux series formally satisfying it, asymptotically approximate of Gevrey order one solutions to this equation in sectors with the vertices at infinity.
A condition sufficient for the convergence of formal solutions of an ODE with analytic left-hand side is well-known.
On the other hand sufficient conditions for the divergence given in the same terms are unknown.
We present the family of values of the parameters such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.
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