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Abstract
In many physical problems, it is important to capture exponentiallysmall effects that lie beyondallorders of a typical asymptotic expansion; when collected, the full expansion is known as the transseries. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a transseries expansion, typically in the context of singular nonlinear perturbative differential or integral equations. Separate to applied exponential asymptotics, there exists a closely related line of development known as Écalle's theory of resurgence, which describes the connection between transseries 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 Écalle's resurgence theory have mainly focused on nonparametric 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 factorialoverpower 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 factorialoverpower 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 lateterm behaviours.
Original language  English 

Pages (fromto)  9741025 
Journal  Studies in Applied Mathematics 
Volume  152 
Issue number  3 
Early online date  26 Dec 2023 
DOIs  
Publication status  Published  30 Apr 2024 
Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.Fingerprint
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 1 Active

Exponential asymptotics for multidimensional systems in fluid mechanics
Trinh, P. (PI)
Engineering and Physical Sciences Research Council
1/04/21 → 31/05/25
Project: Research council