Abstract

In the first half of the talk, I will demonstrate the propagation of nonlinear waves in singularly-perturbed chains of particles with nearest-neighbour interactions (including systems such as diatomic Toda chains and woodpile Hertzian chains). The purpose of this analysis is to calculate exponentially small oscillations that appear due to Stokes’ Phenomenon, arising due to singularities in the analytic continuation of the leading-order solitary wave. As the problems increase in complexity, calculating the leading-order wave becomes impossible. I will demonstrate several methods for approximating the leading-order behaviour, including a method known as the AAA algorithm, and consider the effects of these approximations on the analytic continuation. Can the Stokes structure be recovered?  We will see that the AAA method appears to be capable of reproducing the Stokes switching behaviour correctly, despite having singularity strength that is different to the “true” analytic continuation. I will attempt to explain how this can occur by considering a simple linear differential equation, and show that—for linear problems at least—the method is trustworthy.