Exponential asymptotics and higher-order Stokes phenomenon in singularly perturbed ODEs

The higher-order Stokes phenomenon can emerge in the asymptotic analysis of many problems governed by singular perturbations. Indeed, over the last two decades, the phenomenon has appeared in many physical applications, from acoustic and optical wave phenomena and gravity-capillary ripples to models of crystal growth and equatorial Kelvin waves. It emerges in a generic fashion in the exponential asymptotics of higher-order ordinary and partial differential equations. The intention of this work is to highlight its importance, and develop further practical methodologies for the study of higher-order Stokes phenomena, primarily for general nonintegrable problems. Our formal methodology is demonstrated through application to a second-order linear inhomogeneous ODE that exemplifies the simplest example of higher-order Stokes phenomena. In this model problem, the Borel transform can be derived explicitly, and this gives insight into the beyond-all-orders structure. We review and study additional examples, with physically important connections, including higher-order ODEs and eigenvalue problems.