Appearance of the higher-order Stokes phenomenon in a discrete Airy equation

Abstract

We study a discrete variant of the Airy equation, formulated as an advance-delay equation, to reveal that discretization induces the higher-order Stokes phenomenon, which is not present in the continuous Airy function and is typically only encountered in solutions to third-order or higher linear homogeneous, or nonlinear, differential equations. Using steepest descent and direct series methods, we derive asymptotic solutions and the Stokes structure. Our analysis shows that discretization produces a more intricate Stokes structure, containing higher-order Stokes phenomena and infinite accumulations of Stokes and anti-Stokes curves. The latter feature is a strictly nonlinear effect in continuous differential equations. We show that this unusual behavior can be generated in a discrete equation from a linear discretization. Numerical simulations confirm the predictions, and a direct comparison with the continuous Airy equation explains how the discretization alters the Stokes structure.