Resurgent aspects of applied exponential asymptotics

TL;DR

This paper explores the deep connections between applied exponential asymptotics and Ecalle's resurgence theory, developing a Borel plane framework to enhance understanding of exponentially-small effects in complex problems.

Contribution

It introduces an alternative Borel plane framework for exponential asymptotics, clarifies key heuristic elements, and provides tools for complex asymptotic problems involving singularities and boundary layers.

Findings

Developed a Borel plane-based framework for exponential asymptotics.

Clarified the use of Van Dyke's rule and factorial-over-power ansatzes.

Provided tools for analyzing problems with coalescing singularities and nested boundary layers.

Abstract

In many physical problems, it is important to capture exponentially-small effects that lie beyond-all-orders of a typical asymptotic expansion; when collected, the full expansion is known as the trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically in the context of singular non-linear perturbative differential or integral equations. Separate to applied exponential asymptotics, there exists a closely related line of development known as \'Ecalle's theory of resurgence, which describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. This connection is realised through the process of Borel resummation. However, in contrast to singularly perturbed problems, Borel resummation and \'Ecalle's resurgence theory have mainly focused on non-parametric asymptotic expansions (i.e. differential equations without a parameter). The relationships between these latter areas and applied exponential asymptotics has not been thoroughly examined, partially due to differences in language and emphasis. In this work, we explore these connections by developing an alternative framework for the factorial-over-power ansatz in exponential asymptotics that is centred on the Borel plane. Our work clarifies a number of elements used in applied exponential asymptotics, such as the heuristic use of Van Dyke's rule and the universality of factorial-over-power ansatzes. Along the way, we provide a number of useful tools for probing more pathological problems in exponential asymptotics known to arise in applications; this includes problems with coalescing singularities, nested boundary layers, and more general late-term behaviours.