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25

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

0.6

0.5

0.4

u

0.3

0.2

Wing Core Wing

0.1

}±

0

-20 -10 0 10 20

x

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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27

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 ≡

2j

ajm sech2m

uj , (41)

m=1

j=0

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

xx

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)

by

2

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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29

Exponential Asymptotics

5. Special Numerical Algorithms, especially Spectral Methods

6. Sequence Acceleration including Pad´ and Hermite-Pad´ Approx-

e e

imants

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’∞

(47)

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