Resurgent deformations for an ordinary differential equation of order 2

Eric Delabaere and Jean-Marc Rasoamanana

Vol. 223 (2006), No. 1, 35–93

DOI: 10.2140/pjm.2006.223.35

Abstract

We consider the differential equation (d²∕dx²)Φ(x) = (P_m(x)∕x²)Φ(x) in the complex field, where P_m is a monic polynomial function of order m. We investigate the asymptotic and resurgent properties of the solutions at infinity, focusing in particular on the analytic dependence of the Stokes–Sibuya multipliers on the coefficients of P_m. Taking into account the nontrivial monodromy at the origin, we derive a set of functional equations for the Stokes–Sibuya multipliers, and show how these relations can be used to compute the Stokes multipliers for a class of polynomials P_m. In particular, we obtain conditions for isomonodromic deformations when m = 3.

Keywords

resurgence theory, Stokes phenomena, connection problem

Mathematical Subject Classification 2000

Primary: 34L40, 34M37, 34M40

Milestones

Received: 5 March 2004

Accepted: 18 January 2005

Published: 1 January 2006