Exponentially small corrections to divergent asymptotic expansions of solutions of the fifth Painlevé equation

We calculate the leading term of asymptotics for the coefficients of certain divergent asymptotic expansions for the solutions of the fifth Painlevé equation (P₅) by using the isomonodromy deformation method and the Borel transform. Unexpectedly, these asymptotics appear to be periodic functions of the coefficients of P₅. 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 existence of rational solutions for P₅; (3) and a numerical study of the tau-function for P₅.

Full Text (PDF format)

Published 1 January 1997