An algebraic study of parametric Stokes phenomena

TL;DR

This paper explores the geometric structure of parametric Stokes phenomena using algebraic curves, providing new insights into singularities, turning points, and resurgence in asymptotic analysis.

Contribution

It introduces an algebraic curve approach to analyze parametric resurgence and systematically describes analytic continuation and Stokes constants.

Findings

Characterization of singularity structures and turning points.

Development of a Borel plane inner-outer matching procedure.

Systematic description of Stokes phenomena in algebraic curve context.

Abstract

We investigate geometric aspects of co-equational parametric resurgence, by studying physical problems whose formal asymptotic solutions give rise to Borel transforms lying on an algebraic curve. This perspective allows us to elucidate concepts unique to parametric resurgence such as singularity structures, (virtual) turning points and the higher-order Stokes phenomenon. We construct examples as solutions to Borel plane partial differential equations using an algebraic curve ansatz before turning to the general analytic structure of co-equational resurgence problems, where we provide a systematic description of analytic continuation and Stokes constants through a Borel plane inner-outer matching procedure.