Vol. 223, No. 1, 2006

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Resurgent deformations for an ordinary differential equation of order 2

Eric Delabaere and Jean-Marc Rasoamanana

Vol. 223 (2006), No. 1, 35–93
Abstract

We consider the differential equation (d2dx2)Φ(x) = (Pm(x)x2)Φ(x) in the complex field, where Pm 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 Pm. 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 Pm. 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
Authors
Eric Delabaere
Département de Mathématiques
UMR CNRS 6093
Université d’Angers
2 Boulevard Lavoisier
49045 Angers Cedex 01
France
Jean-Marc Rasoamanana
Département de Mathématiques
UMR CNRS 6093
Université d’Angers
2 Boulevard Lavoisier
49045 Angers Cedex 01
France