Smoothing of the Higher-Order Stokes Phenomenon

Abstract

For over a century, the Stokes phenomenon had been perceived as a discontinuous change in the asymptotic representation of a function. In 1989, Berry demonstrated it is possible to smooth this discontinuity in broad classes of problems with the prefactor for the exponentially small contribution switching on/off taking a universal error function form. Following pioneering work of Berk, Nevins, and Roberts and the Japanese school of formally exact asymptotics, the concept of the higher-order Stokes phenomenon was introduced, 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 Borel-plane singularities transitioning between 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 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. 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 present in the vicinity of a Stokes line, but which rapidly tends to zero on either side. We also include a rigorous derivation 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.