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elementary calculus which are su¬cient to derive divergent series.
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

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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]

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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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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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]

|f ( ) ’ |∼O N +1
aj (1)

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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
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.


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