Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 271

Exponentially Small Corrections to Divergent Asymptotic Expansions of Solutions of the Fifth Painlevé Equation

F. V. Andreev, A. V. Kitaev

We calculate the leading term of asymptotics for the coefficients of certain divergent asymptotic expansions of solutions to the fifth Painlevé equation (P_5) by using the isomonodromy deformation method and the Borel transform. Unexpectedly, these asymptotics appear to be periodic functions of the coefficients of P_5. We also show the relation of our results with some other facts already known in the theory of the Painlevé equations established by other methods: (1) a connection formula for the third Painlevé equation; (2) a condition for the existence of rational solutions of P_5; and (3) a numerical study of the tau-function for P_5.

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