Smoothing of the higher-order Stokes phenomenon

Abstract

For nearly a century and a half the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989 Berry demonstrated how it is possible to smooth out this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution that is being switched on/off taking the universal form of an error function. Following pioneering work of Berk {\it et al.} \cite{BNR82} and the Japanese school of formally exact asymptotics \cite{Aokietal1994,AKT01}, the concept of the higher-order Stokes phenomenon was introduced in \cite{HLO04} and \cite{CM05}, whereby the ability for the exponentially small terms to cause a Stokes phenomenon may change, depending on the values of parameters in the problem, corresponding to the associated singularities in the Borel plane transitioning between different Riemann sheets. Until now, the higher-order Stokes phenomenon has also been treated as a discontinuous event. In this paper we show how the higher-order Stokes phenomenon is, in fact, also smooth and occurs universally with a prefactor that takes the form of a new special function, based on a Gaussian convolution of an error function that gives rise to a rich structure. We provide a rigorous derivation of the result, with examples spanning the gamma function, a second-order nonlinear ODE and the telegraph equation, giving rise to a ghost-like smooth contribution that is present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation and example of the effect of the smoothed higher-order Stokes phenomenon on the individual terms in the asymptotic series, where the additional contributions appear prefactored by an error function.