A. B. Olde Daalhuis

Stokes phenomenon and matched asymptotic expansions

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.

Uniform Airy-type expansions of integrals

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 are 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 approximants, 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.

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.

UNIFORM ASYMPTOTIC EXPANSIONS FOR HYPERGEOMETRIC FUNCTIONS WITH LARGE PARAMETERS III

In this paper, we discuss asymptotic expansions for the Gauss hypergeometric function 2F1(a + e1λ, b + e2λ; c + e3λ; z), where ej = 0, ±1, as |λ| → ∞. We complete the results of two previous publications, extend the sectors of validity, and give more details on the computation of the coefficients.

Inertia-gravity-wave radiation by a sheared vortex

We consider the linear evolution of a localized vortex with Gaussian potential vorticity that is superposed on a horizontal Couette flow in a rapidly rotating strongly stratified fluid. The Rossby number, defined as the ratio of the shear of the Couette flow to the Coriolis frequency, is assumed small. Our focus is on the inertia-gravity waves that are generated spontaneously during the evolution of the vortex. These are exponentially small in the Rossby number and hence are neglected in balanced models such as the quasi-geostrophic model and its higher-order generalizations. We develop an exponential-asymptotic approach, based on an expansion in sheared modes, to give an analytic description of the three-dimensional structure of the inertia-gravity waves emitted by the vortex. This provides an explicit example of the spontaneous radiation of inertia-gravity waves by localized balanced motion in the small-Rossby-number regime. The inertia-gravity waves are emitted as a burst of four wavepackets propagating downstream of the vortex. The approach employed reduces the computation of inertia-gravity-wave fields to a single quadrature, carried out numerically, for each spatial location and each time. This makes it possible to unambiguously define an initial state that is entirely free of inertia-gravity waves, and circumvents the difficulties generally associated with the separation between balanced motion and inertia-gravity waves.

Hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one

In this paper we discuss the special properties of hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one. We show that the re–expansions are independent of the inhomogeneity. We illustrate how this leads to a symmetry breaking in the Stokes constants within a pair of formal solutions of a differential equation. A consequence is that Stokes constants may exactly vanish in higher–order equations, leading to dramatic simplifications in the hyperasymptotic structures. Two examples are included.

Stokes-multiplier expansion in an inhomogeneous differential equation with a small parameter

Accurate approximations to the solutions of a second-order inhomogeneous equation with a small parameter ε are derived using exponential asymptotics. The subdominant homogeneous solutions that are switched on by an inhomogeneous solution through a Stokes phenomenon are computed. The computation relies on a resurgence relation, and it provides the ε-dependent Stokes multiplier in the form of a power series. The ε-dependence of the Stokes multiplier is related to constants of integration that can be chosen arbitrarily in the WKB-type construction of the homogeneous solution. The equation under study governs the evolution of special solutions of the Boussinesq equations for rapidly rotating, strongly stratified fluids. In this context, the switching on of subdominant homogeneous solutions is interpreted as the generation of exponentially small inertia–gravity waves.

INVERSE FACTORIAL-SERIES SOLUTIONS OF DIFFERENCE EQUATIONS

We obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large

UNIFORM ASYMPTOTIC APPROXIMATIONS FOR THE MEIXNER-SOBOLEV POLYNOMIALS

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Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation

asymptotics of the hypergeometric function