Analytic continuation of Borel sum of formal solution of semilinear partial differential equation

Abstract

We study the Borel summability and the analytic behavior of the Borel sum of a formal solution of first-order semilinear system with a singular perturbative parameter. By virtue of the representation formula of the Borel sum of a formal series solution expanded in terms of a parameter, we show that the analytic continuation of the Borel sum with respect to the parameter to a regular point in a singular direction coincides with the solution of the initial value problem expanded in the space variable. We also show that a similar phenomenon occurs outside the origin of the independent variables.