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Exponential Asymptotics

between the boundaries. Here, the boundary-layer-induced error is only
the square of the minimum error in the power series for up (x; ) when
f (x) = 4/(1 + x2 ).
The outer solution is a greater villain. Even without boundaries, the
multiple scales series is divergent.

7. Weakly Nonlocal Solitary Waves

“In general, the divergence of series in perturbation theory (while
a good approximation is given by a few initial terms) is usually
related to the fact that we are looking for an object which does
not exist. If we try to ¬t a phenomenon to a scheme which actually
contradicts the essential features of the phenomenon, then it is not
surprising that our series diverge.”
V. I. Arnold, (1937“)[7], pg. 395.

Solitary waves, which are spatially localized nonlinear disturbances
that propagate without change in shape or form, have been important
in a wide range of science and engineering disciplines. Such diverse
phenomena as the Great Red Spot of Jupiter, Gulf Stream rings in
the ocean, neural impulses, vibrations in polymer lattices, and perhaps
even the elementary particles of physics have been identi¬ed, at least
tentatively, as solitary waves; in ten years, most of our phone and data
communications may be through exchange of envelope solitary waves
in ¬ber optics.
Classic examples of solitary waves decay exponentially fast away
from the peak of the disturbance. In the last few years, as reviewed
in the author™s book [72] and also [56], it has become clear that soli-
tary waves which ¬‚unk the decay condition are equally important. Such
“weakly nonlocal” solitary waves decay not to zero, but to an oscilla-
tion of amplitude ±, the “radiation coe¬cient” (Fig. 5). The amplitude
of these oscillations is important because it determines the radiative
lifetime of the disturbance.
The complication is that for many nonlocal solitary waves, the radi-
ation coe¬cient ± is an exponential function of 1/ where is a small
parameter proportional to the amplitude of the maximum of the soli-
tary wave. This implies that an ordinary asymptotic series in powers
of :

’ must fail to converge to the solution.

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26 John P. Boyd


Wing Core Wing

-20 -10 0 10 20
Figure 5. Schematic of a weakly nonlocal solitary wave or a forced wave of simi-
lar shape. The amplitude of the “wings” is the “radiation coe¬cient” ±, which is
exponentially small in 1/ compared to the amplitude of the“core”

’ must tell us nothing about whether the solitary waves are classical
or weakly nonlocal.
’ must be useless for computing ±.
However, it is possible to compute the radiation coe¬cient through a
hyperasymptotic approximation [68],[72].
A full treatment of a weakly nonlocal soliton is too complicated for
an introduction to hyperasymptotics, but it is possible to give the ¬‚a-
vor of the subject through the closely-related inhomogeneous ordinary
di¬erential equation studied by Akylas and Yang [5]
uxx + u ’
2 22
u = sech2 (x) (39)
To lowest order in , the second derivative is negligible compared to
u, just as in our previous example, and the quadratic term is also small
so that
u(x) ∼ sech2 (x) (40)

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Exponential Asymptotics

By assuming u(x) may be expanded as a power series in even powers
of , substituting the result into the di¬erential equation and matching
powers one ¬nds
∞ j+1
u(x) ∼ uj ≡
ajm sech2m
uj , (41)

When this series is truncated to ¬nite order, j ¤ N , all terms in
the truncation decay exponentially with |x| and therefore so does the
approximation uN . In reality, the exact solution decays to an oscil-
lation, just as in Fig. 5. The “wings” are invisible to the multiple
scales/amplitude expansion because the amplitude ± of the wings is
an exponential function of 1/ .
Boyd shows [68] [with notational di¬erences from this review] that
the residual equation which must be solved at each order is

uN +1 = r(uN ) (42)

where r(uN ) ≡ ’{ 2 uN + uN ’ 2 (uN )2 ’ sech2 (x)} is the ”residual
function” of the solution up to and including N -th order. When the
order N = Noptimum ∼ ’1/2 + π/(4 ), the Fourier transform of the
residual is peaked at wavenumber k = 1/ . In other words, when the
series is truncated at optimal order, the neglected second derivative is
just as important as uN +1 in consistently computing the correction at
next order. The hyperasymptotic approximation is to replace Eq. (42)
uN +1,xx + uN +1 = r(uN ) (43)
for all N > Noptimum .
The good news is that the nonlinear term in the original forced-
KdV equation is still negligible on the left-hand side of the pertur-
bation equations at each order (though it appears in the residual on
the right-hand side). The bad news is that the equation we must solve
to compute the hyperasymptotic corrections, although linear, does not
admit a closed form solution except in the form of an integral which
cannot generally be evaluated analytically:
∞ RN (k)
2 N +1
u (x) = exp(ikx)dk (44)
1 ’ 2 k2

where RN (k) is the Fourier transform of the residual of the N-th order
perturbative approximation.
The Euler expansion cannot help; a weighted sum of the terms of
the original asymptotic series must decay exponentially with | x | and

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28 John P. Boyd

therefore will miss the oscillatory wings. The integrand in Eq. 44 is
now singular on the integration interval, rather than o¬ it as for the
Stieltjes function. Indeed, when N ≈ Noptimum ( ), the numerator of
the integrand is largest at | k |= 1/ , precisely where the denominator
is singular! No simple change in the center of the Taylor expansion of
the denominator factor 1/(1 ’ 2 k 2 ) will help here.
Fortunately, it is possible to partially solve Eq. 43 in the sense that
we can analytically determine the amplitude of the radiation coe¬cient
±. Boyd [68] shows that ± is just the Fourier transform of the residual
at the points of singularity. The result is an approximation to ±( )
with relative error O( 2 ). This can be extrapolated to the limit ’ 0
to obtain
1.558823 + O( 2 ) π
exp ’
±( ) = , << 1 (45)
2 2
As for the Stieltjes integral, several di¬erent hyperasymptotic meth-
ods are available for weakly nonlocal solitary waves and related prob-
lems. The most widely used is to match asymptotic expansions near
the singularities of the solitary wave on the imaginary axis. Originally
developed by Pokrovskii and Khalatnikov [262] for “above-the-barrier”
quantum scattering (WKB theory in the absence of a turning point),
it was ¬rst applied to nonlinear problems by Kruskal and Segur [278],
[172]. The book by Boyd [72] reviews a wide number of applications
and improvements to the PKKS method.
Akylas and Yang[5, 323, 324, 325, 327] apply multiple scales per-
turbation theory in wavenumber space after a Fourier transformation.
Chapman, King and Adams[96], Costin[104, 105] and Costin and Kruskal[106,
107], Ecalle[123] have all shown that related but distinct methods can
also be applied to nonlinear di¬erential equations.

8. Overview of Hyperasymptotic Methods

Hyperasymptotic methods include the following:

1. (Second) Asymptotic Approximation of Error Integral or Residual
Equation for Superasymptotic Approximation

2. Isolation Strategies, or Rewriting the Problem so the Exponentially
Small Thing is the Only Thing

3. Resurgence Schemes or Resummation of Late Terms

4. Complex-Plane Matching of Asymptotic Expansions

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Exponential Asymptotics

5. Special Numerical Algorithms, especially Spectral Methods
6. Sequence Acceleration including Pad´ and Hermite-Pad´ Approx-
e e
7. Hybrid Numerical/Analytical Perturbative Schemes
The labels are suggestive rather than mutually exclusive. As shown
amusingly in Nayfeh [229], the same asymptotic approximation can
often be generated by any of half a dozen di¬erent methods with seem-
ingly very dissimilar strategies. Thus, the Euler summation gives the
exact same sequence of approximations, when applied to the Stieltjes
function, as making a power series expansion in the error integral for
the superasymptotic approximation.
In the next few sections,we shall brie¬‚y discuss each of these general
strategies in turn.

9. Isolation of Exponential Smallness

Long before the present surge of interest in exploring the world of the
exponentially small, some important problems were successfully solved
without bene¬t of any of the strategies of modern hyperasymptotics.
The key idea is isolation: in the region of interest (perhaps after a
transformation or rearrangement of the problem), the exponentially
small quantity is the only quantity so that it is not swamped by other
terms proportional to powers of .
A quantum mechanical example is the “WKB”, “phase-integral” or
“Liouville-Green” calculation of “Below-the-Barrier Wave Transmis-
sion”. The goal is to solve the stationary Schroedinger equation

ψxx + {k 2 ’ V ( x)}ψ = 0 (46)

subject to the boundary conditions of (i) an incoming wave from the
left of unit amplitude and (ii) zero wave incoming from the right:

ψ ∼ exp(ikx)+± exp(’ikx), x ’ ’∞; ψ ∼ β exp(ikx),x’∞
The goal is to compute the amplitudes of the re¬‚ected and transmit-
ted waves, ± and β, respectively. If k 2 < max(V ( x)),however, β is
exponentially small in 1/ for ¬xed k, and ± di¬ers from unity by an
exponentially small amount also. Nevertheless, this problem was solved
in the 1920™s as reviewed in Nayfeh [229] and Bender and Orszag [19].
The crucial point is that on the right side of the potential barrier,
the exponentially small transmitted wave is the entire wavefunction.

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30 John P. Boyd

Pumped Flow


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