On the higher–order Stokes phenomenon

During the course of a Stokes phenomenon, an asymptotic expansion can change its form as a further series, prefactored by an exponentially small term and a Stokes multiplier, appears in the representation. The initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters. In this paper we introduce the concept of a‘higher–order Stokes phenomeno’, at which a Stokes multiplier itself can change value. We show that the higher–order Stokes phenomenon can be used to explain the apparent sudden birth of Stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions. We provide an example of how the higher–order Stokes phenomenon can have important effects on the large–time behaviour of partial differential equations.