In the next section, we begin with a brief catalogue of physics,

chemistry and engineering problems where key parts of the answer lie

“beyond all orders” in the standard asymptotic expansion because these

features are exponentially small in 1/ where << 1 is the perturba-

tion parameter. The emerging ¬eld of “exponential asymptotics” is not

a branch of pure mathematics in pursuit of beauty (though some of the

ideas are aesthetically charming) but a matter of bloody and unyielding

engineering necessity.

In Sec. 3, we review some concepts that are already scattered in the

textbooks: Poincar´™s de¬nition of asymptoticity, optimal truncation

e

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6 John P. Boyd

Table III. Weakly Nonlocal Solitary Waves

Species Field References

Capillary-gravity Oceanography, Pomeau et al. [263]

Water Waves Marine Engineering Hunter & Scheurle [153]

Boyd [62]

Benilov, Grimshaw

& Kuznetsova [22]

Grimshaw & Joshi [134]

Dias et al [114]

φ4 Breather Particle Physics Segur & Kruskal [278]

Boyd [58]

Fluxons, DNA helix Physics Malomed[195]

Modons in Plasma Physics Meiss & Horton [201]

Magnetic Shear

Klein-Gordon Electrical Boyd [67]

Envelope Solitons Engineering Kivshar&Malomed[167]

Various Review article Kivshar&Malomed[168]

Higher Latitudinal Oceanography Boyd [56, 57]

Mode Rossby Waves

Higher Vertical Oceanography, Akylas & Grimshaw [4]

Mode Internal Marine

Gravity Waves Engineering

Perturbed Physics Malomed[194]

Sine-Gordon

Nonlinear Schr¨dinger

o Nonlinear Optics Wai, Chen & Lee[307]

Eq., Cubic Dispersion

Self-Induced Nonlinear Optics Branis, Martin & Birman [84]

Transparency Equs: Martin & Branis [197]

Envelope Solitons

Internal Waves: Oceanography, Vanden-Broeck & Turner [299]

Strati¬ed Layer Marine

Between 2 Constant Engineering

Density Layers

Lee waves Oceanography Yang & Akylas [325]

Pseudospectra of Applied Math., Reddy, Schmid&Henningson [267]

Matrices & Fluid Mechanics Reichel&Trefethen [268]

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7

Exponential Asymptotics

and minimum error, Carrier™s Rule, and four heuristics for predicting

divergence: the Exponential Reciprocal Rule, Van Dyke™s Principle of

Multiple Scales, Dyson™s Change“of“Sign Argument, and the Principle

of Non-Uniform Smallness. In later sections, we illustrate hyperasymp-

totic perturbation theory, which allows us to partially overcome the

evils of divergence, through three examples: the Stieltjes function (Secs.

4 and 5), a linear inhomogeneous di¬erentiation equation (Sec. 6), and

a weakly nonlocal solitary wave (Sec. 7).

Lastly, in Sec. 8 we present an overview of hyperasymptotic methods

in general. We use the Pokroskii-Khalatnikov-Kruskal-Segur (PKKS)

method for “above-the-barrier” quantum scattering (Sec. 14) and ”resur-

gence” for the analysis of Stokes™ phenomenon (Secs. 12 and 13) to give

the ¬‚avor of these new ideas. (We warn the reader: “beyond all orders”

perturbation theory has become su¬ciently developed that it is impos-

sible, short of a book, to even summarize all the useful strategies.) The

¬nal section is a summary with pointers to further reading.

2. The Necessity of Computing Exponentially Small Terms

“Even the best toolmaker cannot wring ¬ve-¬gure accuracy out of

the machining tolerances... This is how I come to ¬nd nearly all

computations to more than three signi¬cant ¬gures embarrassing.

It™s not a criticism of computer science because there is a direct

analogy in asymptotic expansions. I ¬nd them plain embarrassing

as a failure of realistic judgment.”

“I was led to contemplate a heretical question: are higher approxi-

mations than the ¬rst justi¬able? My experience indicates yes, but

rarely. All di¬erential equations are imperfect models and I would be

embarrassed to publish a second approximation without convincing

justi¬cation that the quality of the model validates it.”

“Solutions as an end in themselves are pure mathematics; do we

really need to know them to eight signi¬cant decimals? ”

” Richard E. Meyer (1992) [218]

Meyers™ tart comment illuminates a fundamental limitation of hyper-

asymptotic perturbation theory: for many engineering and physics appli-

cations, a single term of an asymptotic series is su¬cient. When more

than one is needed, this usually means that the small parameter is

not really small. Hyperasypmtotic methods depend, as much as con-

ventional perturbation theory, on the true and genuine smallness of

and so cannot help. Numerical algorithms are usually necessary when

∼ O(1), either numerical or analytic [63].

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8 John P. Boyd

And so, the ¬rst question of any adventure in hyperasymptotics is

a question that patriotric Americans were supposed to ask themselves

during wartime gas-rationing: “Is this trip necessary?” The point of

this review is that there is an amazing variety of problems where the

trip is necessary.

Table I is a collection of miscellaneous problems from a variety of

¬elds, especially ¬‚uid mechanics, where exponential smallness is cru-

cial. Tables II and III are restricted selections limited to two areas

where “beyond all orders” calculations have been especially common:

quantum mechanics and the weakly nonlocal solitary waves. The com-

mon thread is that for all these problems, some aspect of the physics

is exponentially small in 1/ where is the perturbation parameter.

Since exp(’q/ ) where q is a constant cannot be approximated as a

power series in “ all its derivatives are zero at = 0 “ such exponen-

tially small e¬ects are invisible to an power series. Such “beyond all

orders” features are like mathematical stealth aircraft, ¬‚ying unseen by

the radar of conventional asymptotics.

There are several reasons why such apparently tiny and insigni¬cant

features are important. In quantum chemistry and physics, for exam-

ple, perturbations such as an external electric ¬eld may destabilize

molecules. Mathematically, the eigenvalue E of the Schroedinger equa-

tion acquires an imaginary part which is typically exponentially small

in 1/ . Nevertheless, this tiny (E) is important because it completely

controls the lifetime of the molecule. J. R. Oppenheimer [255] showed

that in the presence of an external electric ¬eld of strength , hydrogen

atoms disassociated on a timescale which is inversely proportional to

(E) = (4/3 ) exp(’2/(3 )) and that electrons can be similarly sprung

from metals. (This observation was the basis for the development of the

scanning tunneling microscope by Binnig and Rohrer half a century lat-

er.) Only a few months after Oppenheimer™s 1928 article, G. Gamow

and Condon and Gurney showed that this “tunnelling” explained the

radioactive decay of unstable nuclei and particles, again on a timescale

exponentially small in the reciprocal of the perturbation parameter.

Similarly, weakly nonlocal solitary waves do not decay to zero as

| x |’ ∞ but to small, quasi-sinusoidal oscillations that ¬ll all of

space. For the species listed in Table III, the amplitude of the “radia-

tion coe¬cient” ± is proportional to exp(’q/ ) for some q. When the

appropriate wave equations are given a spatially localized initial con-

dition, the resulting coherent structure slowly decays by radiation on

a timescale inversely proportional to ±.

For other problems, exponential smallness may hold the key to the

very existence of solutions. For example, the melt interface between a

solid and liquid is unstable, breaking up into dendritic ¬ngers. Ivant-

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9

Exponential Asymptotics

sev (1947) develped a theory that successfully explained the parabolic

shape of the ¬ngers. However, experiments showed that the ¬ngers also

had a de¬nite width. Attempts to predict this width by a power series in

the surface tension failed miserably, even when carried to high order.

Eventually, it was realized that the instability is controlled by factors

that lay beyond all orders in . Kruskal and Segur [171, 172] showed

that the complex-plane matched asymptotics method of Pokrovskii and

Khalatnikov [262] could be applied to a simple model of crystal growth.

In so doing, they not only resolved a forty-year old conundrum, but also

furnished one of the (multiple) triggers for the resurgence in exponen-

tial asymptotics.

Even earlier, the ¬‚ow of laminar ¬‚uid through a pipe or channel

with porous walls had been shown to depend on exponential small-

ness. This nonlinear ¬‚ow is not unique; rather there are two solutions

which di¬er only through terms which are exponentially small in the

Reynolds number R, which is the reciprocal of the perturbation param-

eter . As early as 1969, Terrill [292, 291] had diagnosed the illness and

analytically determined the exponentially-small, mode-splitting terms

[272, 135]

Similarly, the interactions between the electrostatic ¬elds of atoms

cause splitting of molecular spectra. The prototype is the quantum

+

mechanical “double well”, such as the H2 ion. The eigenvalues of the

Schroedinger equation come in pairs, each pair close to the energy of

an orbital of the hydrogen atom. The di¬erence between each pair is

exponentially small in the internuclear separation.

Lastly, Stokes™ phenomenon in asymptotic expansions, which requires

one exponential times a power series in in regions of the complex -

plane, but two exponentials in other sectors, can only be smoothed and

fully understood by looking at exponentially small terms.

In the physical sciences, smallness is relative. We can no more auto-

matically assume an e¬ect is negligible because it is proportional to

exp(’q/ ) than a mother can regard her baby as insigni¬cant because

it is only sixty centimeters long.

3. De¬nitions and Heuristics

De¬nition 1 (Asymptoticity). A power series is asymptotic to a

function f ( ) if, for ¬xed N and su¬ciently small [19]

N

|f ( ) ’ |∼O N +1

j

aj (1)

j=0

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10 John P. Boyd

where O() is the usual “Landau gauge” symbol that denotes that the

quantity to the left of the asymptotic equality is bounded in absolute

value by a constant times the function inside the parentheses on the

right. This formal de¬nition, due to Poincar´, tells us what happens

e

in the limit that tends to 0 for ¬xed N . Unfortunately, the more

interesting limit is ¬xed, N ’ ∞. A series may be asymptotic, and

yet diverge in the sense that for su¬ciently large j, the terms increase

with increasing j.

However, convergence may be over-rated as expressed by the follow-

ing amusing heuristic.

Proposition 1 (Carrier™s Rule). “Divergent series converge faster

than convergent series because they don™t have to converge.”

What George F. Carrier meant by this bit of apparent jabberwocky

is that the leading term in a divergent series is often a very good approx-

imation even when the “small” parameter is not particularly small.

This is illustrated through many numerical comparisons in [19]. In con-

trast, it is quite unusual for an ordinary convergent power series to be

accurate when ∼ O(1).

The vice of divergence is that for ¬xed , the error in a divergent

series will reach, as more terms are added, an “dependent minimum.

The error then increases without bound as the number of terms tends

to in¬nity. The standard empirical strategy for achieving this minimum

error is the following.

De¬nition 2 (Optimal Truncation Rule). For a given , the min-

imum error in an asymptotic series is usually achieved by truncating

the series so as to retain the smallest term in the series, discarding all

terms of higher degree.

The imprecise adjective “usually” indicates that this rule is empir-

ical, not something that has been rigorously proved to apply to all

asymptotic series. (Indeed, it is easy to contrive counter-examples.)

Nevertheless, the Optimal Truncation Rule is very useful in practice.