of and the necessary numerics is the full solution to a PDE, but still,

their work is real progress after two decades of no advance at all.

It seems likely that such hybrid numerical-asympotic methods will

¬‚ourish in the next few years for following reasons:

’ Analytical perturbation theory has enjoyed at least a century of

development, and it is hard to grow good ideas in such old soil.

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.82

83

Exponential Asymptotics

’ Mathematics departments, even in the non-computational areas,

are becoming more computer-friendly.

’ Hyrid algorithms have successfully attacked a number of problems

already.

’ There are broad areas where hybrids have not yet been tried.

21. History

Improving upon the minimum error of an asymptotic series has a long

history; Stieltjes himself discussed the possibility in his 1888 doctoral

thesis. Oppenheimer™s calculation of the exponentially large decay time

in the quantum Stark e¬ect and the independent discovery of quantum

tunnelling by Gamow and by Condon and Gurney all happened in 1928.

The Euler acceleration of the Stieltjes function series was ¬rst analyzed

by Rosser [273] in 1951.

One can distinguish several parallel lines of development. The ¬rst is

the calculation of “converging factors” or terminants for the asymptotic

expansions of special functions, beginning with Airey in 1937 [3] and

reaching a high degree of sophistication in the books of Dingle (1973)

and Olver(1974) [118, 249], who also give good histories of earlier work.

Another was quantum mechanics, beginning with discovery of tun-

nelling in 1928, continuing with the Pokrovskii-Khalatnikov solution for

“above-the-barrier” quantum scattering, and continued to the present

with studies of high order perturbation theory. The books written by

Arteca, Fernandez and Castro [8] and edited by LeGuillou and Zinn-

Justin[181] and Braaksma[83] are good testaments, as is a special issue

of International Journal of Quantum Chemistry[269].

A third area is KAM theory and dynamical systems theory in gener-

al. Under perturbations, integrable dynamical systems become chaot-

ic, but the chaos is con¬ned to exponentially thin regions around the

separatrices [136] for small . Through “Arnold di¬usion”, dynamical

systems can move great distances in phase space (on exponentially long

time scales) even when the perturbation is very weak.

A fourth area is “weakly nonlocal solitary waves”, that is, nonlin-

ear coherent structures that would be immortal were it not for weak

radiation away from the core of the structure. These seem to be as

ubiquitous as classical, decaying-to-zero solitary waves. Nonlocal soli-

tary waves arise in ¬ber optics, hydrodynamics, plasmas and a wide

variety of other applications. Meiss and Horton (1983) [201] seem to

have done the earliest explicit calculations. However, the existence of

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.83

84 John P. Boyd

slowly radiating solitary waves in particle physics (“ φ4 breathers”) and

oceanography (Gulf Stream Rings) was known from observations and

initial value computations a decade earlier. The subsequent eruption of

activity is catalogued in the book by Boyd [72].

A ¬fth area is crystal formation and solidi¬cation. The 1985 work

of Kruskal and Segur [171, 172] resolved a long-standing roadblock in

the theory of dendritic ¬ngers on melt interfaces, and touched o¬ a

great plume of activity. There was rapid cross-fertilization with nonlo-

cal solitary waves because Segur and Kruskal applied their new PKKS

method to the “φ4 breather” of particle physics, contributing to the

rapid growth of exponential asymptotics for nonlinear waves.

A sixth area is ¬‚uid mechanics. The Berman-Terrill-Robinson prob-

lem [135] in ¬‚ows with suction, the radiative decay of free oscillations

bound to islands [185] and Kelvin wave instability in oceanography

and atmospheric dynamics [74, 75] were all examples in which expo-

nential smallness had been calculated in the seventies or early eighties.

Somehow, these problems remained isolated. However, boundary layer

theory always involves divergent power series and exponential smallness

as showed by example above. Fluids is an area where hyperasymptotic

technology is likely to have a vigorous future.

A seventh line of research is that pursued by Richard E. Mey-

er and his students. This began with studies of adiabatic invariants

[202, 203, 204, 219, 205, 207]. He also devised an independent solu-

tion to “above the barrier” quantum scattering: recasting the problem

as an integral equation so that the re¬‚ection coe¬cient appears as

the dominant contribution instead of as an exponentially small cor-

rection [206, 208]. This led to further studies of exponential smallness

in water waves trapped around an island [185, 209, 220], connection

across WKB turning points and wave dynamics and quantum tun-

nelling [221, 211, 222, 223, 224, 212, 214, 215, 216, 225, 218, 226].

Meyer has also written four reviews [210, 213, 217, 218].

An eighth line of development is the abstract theory of resurgence

´

and multisummability. This began with Ecalle[123] and continued with

important contributions from Pham, Ramis, Delabaere, Braaksma and

others too numerous to mention as reviewed in [285, 83, 15].

Lastly, a ninth area is the development of resurgence and Stokes

phenomenon by physicists and applied mathematicians. This grew out

´

of the abstract theory of Ecalle, which Berry learned during a visit to

France, but took resurgence in a direction that was less rigorous but

much more pragmatic and applied. The trigger was Berry™s 1989 real-

ization that the discontinuity in the numerical value of an asymptotic

expansion at a Stokes™ line could be smoothed. (The change in topology

of the steepest descent path at a Stokes lines is unavoidable, howev-

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.84

85

Exponential Asymptotics

´

er.) Building on the books of Dingle, Olver and Ecalle, Berry, Howls,

Olver, Olde Daalhuis, Paris, Wood, W. G. C. Boyd and others have

developed smoothed, high order hyperasymptotic approximations for

many species of special functions, for the WKB method and for other

schemes for di¬erential equations. A selection is given in Table IV.

Dingle™s ideas of generic forms for the late terms in asymptotic series

and universal terminants now seem as important to the rise of exponen-

tial asymptotics as the comet-crash(?) that put an end to the dinosaurs

was in biology. Only a year later, Olver™s book developed similar ideas,

with error bounds, for ordinary di¬erential equations. And yet, though

these books were widely bought and read, their net e¬ect at the time

was as quiet as a sandcastle washed away by a rising tide. Bothered by

long-term illness, Dingle never published again.

In recent years, however, the analysis of exponentially small terms

has exploded. A special program of study at the Newton Institute at

Cambridge has brought together researchers from a wide range of ¬elds

for a workshop lasting the whole ¬rst half of 1995. The books by Boyd

[72] and Segur, Tanveer and Levine (eds.) [279] are good introductions

to the vigour and diversity of this interest.

Why was this revolution in asymptotics so slow, so long delayed?

Perhaps the most important factor is that alterations in scienti¬c world-

view, like atom bombs, require assembling a critical mass. Part of this

critical mass was provided by the parallel threads of slow develop-

ment outlined above; when ideas began to cross disciplinary boundaries,

exponential asymptotics exponentiated. Another trigger was the pop-

ularization of algebraic manipulation languages, which made it easier

to compute many terms of an asymptotic series. Lastly, applied math-

ematics is subject to fads and enthusiasms.

I myself read both the Dingle and Olver books when they ¬rst

appeared while I was still in graduate school, but was unimpressed.

First, my was not very small. Second, a string of messy hyperasymp-

totic corrections seemed a poor alternative to numerical algorithms,

which were fast and e¬cient even a quarter century ago. Modern expo-

nential asymptotics still shares these limitations, but there is now a

cadre of enthusiasts who are unbothered as there was not in Dingle™s

time.6

Still, with the emergence of exponential asymptotics as a sub¬eld

of its own with ideas shared widely from physics to ¬‚uids to nonlin-

ear optics, hyperasymptotics has been very useful, at least as the low-

est hyperasymptotic order, in a wide variety of practical applications.

6

In a language of Papua New Guinea, the word “mokita” is used to denote

“things we all know but agree not to talk about”.

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.85

86 John P. Boyd

When the parallel threads ceased to be parallel and converged, the

ancient topic of asymptotics suddenly became very interesting again.

22. Books and Review Articles

The theme of extending asymptotic series through Borel summation

and other methods of re-expanding remainder integrals is treated in the

classic books of Dingle[118] and Olver[249]. Jones™ 1997 book is very

short (160 pages), a primer of steepest descent and hyperasymptotics

that is perhaps closest in style and spirit to the Dingle and Olver books,

but at a somewhat more elementary level. It includes a short appendix

on non-standard analysis as well as exercises at the end of each chapter.

´

Ecalle™s 1981 three-volume treatise greatly extended and generalized

earlier ideas on hyperasymptotics. Unfortunately, his work has not been

translated from French. However, Sternin and Shatalov is a recent pre-

sentation of the abstract theory of resurgence[285]. The collection of

articles edited by Braaksma[83] gives a broader but less coherent state

of the abstract resurgence work. Balser[15] is only one hundred pages

long, but is very readable, based on a course taught by the author.

Kowalenko et al.[169] is a short monograph devoted entirely to the

hyperasymptotics of a fairly narrow class of integrals. Maslov[199] is a

broad treatment of the WKB method.

Segur, Tanveer and Levine[279] is a collection of articles from a

NATO Workshop that displays the remarkable breadth of application

of beyond-all-orders asymptotics that existed even in 1991. Arteca, Fer-

nandez and Castro[8] and LeGuillou and Zinn-Justin [181] describe the

calculation of exponentially small terms in quantum mechanics through

large order perturbation theory and summation methods. Boyd[72] is

focused particularly on nonlocal solitary waves, but it includes a chap-

ter on general applications of hyperasymptotics and several chapters

on numerical methods.

Curiously, review articles seem rarer than books. Berry and Howls[39],

Paris and Wood[260] and Wood[320] have written short, semi-popular

reviews. Delabaere[112] has written (in English) an introduction to

´

Ecalle alient calculus. Olde Daalhuis and Olver[248] describe hyper-

asymptotics (and numerical methods) for linear di¬erential equations.

Byatt-Smith[87],based on an unpublished but widely circulated manuscript

of seven years earlier, is not technically a review, but it nonetheless is

one of the most readable treatments of re-expansion of remainder inte-

grals and the error function smoothing of Stokes phenomenon.

ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.86

87

Exponential Asymptotics

This profusion of books and reviews is helpful, but there are still

some large gaps. This present article was written to ¬ll in some of

these holes and point the reader to other summaries of progress.

23. Summary

“What they [engineers] want from applied mathematics . . . is infor-

mation that illuminates.”

” Richard E. Meyer (1992) [218][pg. 43]

Key concepts:

’ Divergence is a disease caused by a perturbative approximation

which is true for only part of the interval of integration or part of

the Fourier spectrum.

’ A power series is asymptotic when the perturbative assumption is

bad only for a part of the spectrum or integrand that makes an

exponentially small contribution.

’ When a factorially divergent series is truncated at its smallest

term, this “optimal truncation” gives an error which is typically

an exponential function of 1/ . The usual Poincar´ de¬nition of

e

asymptoticity, which refers only to powers of , is therefore rather

misleading. The neologism “superasymptotic” was therefore coined

by Berry and Howls to describe the error in an optimally-truncated

asymptotic series.

’ By appending one or more terms of a second asymptotic series

(with a di¬erent rationale) to the optimal truncation of a divergent

series, one can reduce the error below that of the superasymptot-

ic approximation to obtain a “hyperasymptotic” approximation.

This, too, is divergent, but with a minimum error far smaller than

the best “superasymptotic” approximation. (This rescale-and-add-

another-series step can be repeated for further error reduction.)

’ There are many di¬erent species of hyperasymptotic methods includ-

ing:

1. sequence acceleration schemes such as the Euler, Pad´ and

e

Hermite-Pad´ (Shafer) approximations.