Stokes' Phenomenon and Numerical Analytic Continuation

Date: Thu 25th July 2024

Abstract

Stokes' phenomenon is an asymptotic effect in singularly-perturbed systems that can't be seen using classical power series methods. I will provide a survey of what it is, why it happens, and some physical contexts in which it plays an important role. I will then explain how Stokes' phenomenon is studied using methods known as “exponential asymptotics”.

These methods have been used to study nonlinear waves in particle chains, and to explain the appearance of generalised solitary waves. I will explain why “standard” exponential asymptotics is effective in studying certain particle chains, but also the limitations that restrict its usefulness in others (including the physically significant case of Hertzian chains).

I will show that these limitations can be overcome using numerical analytic continuation, such as the AAA method, to study the structure of complex-plane singularities in the underlying nonlinear waves. By combining this with exponential asymptotics, I will calculate the nonlinear wave behaviour seen in challenging Hertzian particle systems, and explain why this hybrid method substantially increases the utility of exponential asymptotics.