Abstract. This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions.

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 phenomenon', 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.

The formation of shocks in waves of advance in nonlinear partial differential equations is a well-explored problem and has been studied using many different techniques. In this paper we demonstrate how an exponential-asymptotic approach can be used to completely characterize the shock formation in a nonlinear partial differential equation and so resolve an apparent paradox concerning the asymptotic modelling of shock formation. In so doing, we find that the recently discovered higher-order Stokes phenomenon plays a significant, previously unrealized, role in the asymptotic analysis of shocks. For the purposes of clarity, Burgers' equation is used as a pedagogical example, but the techniques illustrated are more generally applicable.

A sequence of re-expansions is developed for the remainder terms in the well-known Poincaré series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one. These re-expansions are a series whose terms are a product of Stokes multipliers, coefficients of the original Poincaré series expansions, and certain multiple integrals, the so-called hyperterminants. Each step of the process reduces the estimate of the error term by an exponentially small factor.The method of this paper is based on the Borel-Laplace transform, which makes it applicable to other problems. At the end of the paper the method is applied to integrals with saddles.Also, a powerful new method is presented to compute the Stokes multipliers. A numerical example is included.

Abstract. A new method for representing the remainder and coefficients in Airy-type expansions of integrals is given. The quantities are written in terms of Cauchy-type integrals and a.re natural generalizations of integral representations of Taylor coefficients and remainders of analytic functions. The new approach gives a general method for extending the domain of the saddle-point parameter to unbounded domains. As a side result the conditions under which the Airy-type asymptotic expansion has a double asymptotic property become clear. An example relating to Laguerre polynomials is worked out in detail. How to apply the method to other types of uniform expansions, for example, to an expansion with Bessel functions as approxima.nts, is explained. In this case the domain of validity can be extended to unbounded domains and the double asymptotic property can be established as well.