Exponential asymptotics, transseries, and generalized Borel summation for analytic rank one systems of ODE's

Abstract

For analytic nonlinear systems of ordinary differential equations, under some non-degeneracy and integrability conditions we prove that the formal exponential series solutions (trans-series) at an irregular singularity of rank one are Borel summable (in a sense similar to that of Ecalle). The functions obtained by re-summation of the trans-series are precisely the solutions of the differential equation that decay in a specified sector in the complex plane. We find the dependence of the correspondence between the solutions of the differential equation and trans-series as the ray in the complex plane changes (local Stokes phenomenon). We study, in addition, the general solution in $\lloc$ of the convolution equations corresponding, by inverse Laplace transform, to the given system of ODE's, and its analytic properties. Simple analytic identities lead to ``resurgence'' relations and to an averaging formula having, in addition to the properties of the medianization of Ecalle, the property of preserving exponential growth at infinity.