6. Continue the matched outer expansion back to the real x-axis to

compute the (exponentially small) magnitude of the Stokes jump

for real x.

The domains of the “inner” and ”outer” regions are illustrated in

Fig. 12.

Step one has already been accomplished by the speci¬cation of the

problem: the relevant critical points are the double poles of f ( x) at

x = ±i/ where the change of variable from x to x has reduced the

residues to 1/ 2 . The shifted coordinate (for matching in the upper

half-plane) is

y ≡ x ’ i/ (93)

Step two pivots on the observation that in the vicinity of its double

pole, it is a legitimate approximation to replace f ( x) by the singular

term only, even though this is a poor approximation everywhere except

near the pole. The inner problem is then

U≡

Uyy + U = 1/y 2 ; 2

u (94)

Step three, computing an outer expansion for the inner problem, is

obtained by an inverse power series in y:

∞

(’1)j+1 (2j ’ 1)!

U (y) ∼ (95)

y 2j

j=1

For the inner problem to be sensible, | y |<< 1/ . For the inverse

power series to be an accurate approximation to the inner solution, we

must have | y |>> 1. It follows that the inverse power series is a good

approximation only in the annulus

1 <<| y |<< 1/ (96)

It is fair to dub this annulus the “matching region” because it will turn

out that the inner limit of the outer expansion will also be legitimate in

this annulus. However, “annulus” is a slightly misleading label because

Eq.(96) ignores Stokes phenomenon, which will limit the validity of

Eq.(95) to a sector of the annulus.

To sort out Stokes phenomenon, it is helpful to sum the divergent

series by Borel summation. For this simple case, the Borel transform

can be written in closed form to give, without approximation,

∞

Bo(y) ≡ exp(’t) log{1 + t2 /y 2 } dt

U (y) = (1/2)Bo(y);

0

(97)

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55

Exponential Asymptotics

The integrand is logarithmically singular at t = ±iy. As the argument

of y varies from 0 to ’π, that is, through a semicircle in the lower half

of the y-plane, the singularity initially in the upper half of the t-plane

rotates clockwise through a semicircle in the right half of the t-plane to

exchange places with the other branch point. As arg(y) passes through

’π/2, that is, through the negative imaginary y ’ axis, the branch

points of the “Borel-logarithm” function Bo(y) are forced to cross the

real t-axis. To avoid discontinuously rede¬ning the branch points of the

logarithm in the integrand, the path of integration must be deformed

to pass below the real t-axis (in the right half t-plane). This gives an

extra contribution which is the Stokes jump for this function with the

negative imaginary y-axis as the Stokes line. One ¬nds

Bo(y) ’ Bo(’y) = 2πi exp(’iy) (98)

The positive and negative real y-axis are the anti-Stokes lines for Bo(y).

The outer expansion is the same as the multiple scales series for

Eq.(26) except for alternating signs; The ¬nal result on the real x-axis

is

±

∞ 2j d2j f ,

’ π exp ’ x≥0

xs j

+ j=0 (’1)

2 dX 2j

u(x) ∼ (99)

∞ d2j f

exp ’ xs

π 2j

j

+ j=0 (’1) , x<0

2 dX 2j

where the outer expansion has been written in terms of derivatives of

f ( x) ≡ f (X) with respect to the “slow” variable X ≡ x to explicitly,

rather than implicitly, display the dependence of the j-th term on 2j .

Extrapolating back to the real axis reduces the magnitude of the

jump by exp(’xs / ) = exp(’1/ ) where xs / is simply the distance of

the singularity from the real x-axis. Note that the exponential depen-

dence on is controlled entirely by xs / ; the strength of the residue

and the type of singularity (simple pole, double pole or logarithm)

only alters factors that vary as powers of or slower.

We chose this particular example because the theory of Pomeau,

Ramani and Grammaticos [263] for the Fifth-Order Korteweg-deVries

equation, later extended to higher order by Grimshaw and Joshi [134],

is very similar. In particular, the dominant singularities “ in their case,

of the lowest order approximation to the solitary wave “ are also dou-

ble poles on the imaginary axis. It is also true that the outer limit

of the inner solution is the Borel-logarithm function, Bo(y) [to lowest

order]. Consequently, the lowest order theory for this nonlinear eigen-

value problem is almost identical to that for this linear, inhomogeneous

problem. The major di¬erence is that the nonlinearity multiplies Bo(y)

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56 John P. Boyd

by a constant which can only be determined numerically by extrapo-

lating the recurrence relation. The early terms of the series in inverse

powers of y in the matching region is strongly a¬ected by the nonlinear-

ity, but the coe¬cients asymptote to those of Bo(y), another triumph

of Dingle™s maxim: Always look at the late terms where a whole class

of problems asymptote to the same, common form.

As noted by a reviewer, the integral for Bo can be integrated by

parts to express it as the sum of two Dingle terminants, and the connec-

tion formulae can then be evaluated through residues. This alternative

derivation of the same answer emphasizes the remarkable universality

of hyperasymptotics; again and again, one keeps falling over the same

small set of terminants.

Table III records many successes for the PKKS method, but it is a

curious success. It is a general truth that the exponential dependence

on is controlled entirely by xs , the distance from the relevant singu-

larities or critical points to the real axis. This is usually almost trivial

to determine. Roughly 90% of the work of the PKKS method is in

determining the “prefactor”, that is, the product of a constant times

algebraic factors of , such as logarithms and powers, which multiplies

the exponential. Not only is the determination of the prefactor (com-

paratively) arduous, but the ¬nal step of determining the overall multi-

plicative constant must always be done numerically. Pomeau, Ramani

and Grammaticos and later workers such as Akylas and Yang [5] and

Boyd [68] have simpli¬ed the numerical bit to extrapolating a sequence

derived from a recurrence, a task much easier than directly solving a

di¬erential equation. However, the fact that the PKKS method is an

analytical method that is not entirely analytic gives much ground to

alternatives such as spectral methods which are discussed later.

15. Snares and Worries: Remote but Dominant Saddle

Points, Ghosts, Interval-Extension and Sensitivity

“There are, so to speak, in the mathematical country, precipices and

pit-shafts down which it would be possible to fall, but that need not

deter us from walking about.”

” Lewis F. Richardson (1925)

More subtle perils in deriving even the lowest order correctly also

lurk. Balian, Parisi and Voros [11] describe an integral where the con-

vergence is controlled by a saddle point at t = 2, but the error is

dominated by the exponentially larger contributions of another saddle

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57

Exponential Asymptotics

point at t = 3. Their function is

∞

I(z) ≡ exp ’z 36t2 ’ 20t3 + 3t4 dt (100)

’∞

For large z, the integrand is steeply peaked about the dominant saddle

point at t = 0 (Fig. 13). The contributions of the other two saddle

points will be proportional to the integrand evaluated at these saddle

points:

exp(’zφ(t = 2)) = exp(’32z), exp(’zφ(t = 3)) = exp(’27z) (101)

Because the saddle point at t = 2 controls convergence, the smallest

term in the asymptotic series for a given z will be O(exp(’32z)), so we

would expect this to be the magnitude of the error in the optimally-

truncated series in inverse powers of z. In reality, the superasymptotic

error is dominated by the contribution of the saddle point at t = 3,

which is O(exp(’27z)) and therefore larger than the smallest term in

the optimally- truncated series by O(exp(5z)).

One of the charms of resurgence theory is that during the early

stages, all saddle points are treated equally. This “saddle point democ-

racy” is valuable in detecting such pathologies, and correctly retain-

ing the contributions of all the important saddle points. Still, if the

asymptotic series is derived not from an integral but directly from a

di¬erential equation so that no information is available but the coe¬-

cients of the series, it would be easy to be fooled, and assume that the

magnitude of the smallest retained term was a genuine estimate of the

superasymptotic error.

Fortunately, it appears that this is rare in practice. The applied

mathematical landscape is littered with deep sinkholes which fortu-

nately have an area of measure zero. The Balian-Parisi-Voros example

was contrived by its authors rather than derived from a real application.

However, related di¬culties are not contrived.

For the so-called φ4 breather problem [278, 58], the convergence of

the divergent series is controlled by the constant in the Fourier series

√

with an expected minimum error of O(exp(’ 2π/(2 ))). However, the

far ¬eld √radiation has a magnitude ± which has been shown to be

O(exp(’ 6π/(2 ))). Thus, after the -power series has been truncated

at optimal order and subtracted from the solution, the correction is

still exponentially large in relative to the weakly nonlocal radiation.

Complex-plane matched asymptotics is not inconvenienced [278], but

the hyperasymptotic method of Boyd [67] would likely fail.

Another danger is illustrated by the function

f ( ) ≡ S( ) + exp(’(1/2)/ ) (102)

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58 John P. Boyd

Balian-Parisi-Voros Example

0

10

Error-

-5

10

Determining

exp(- phi(t))

Saddle Pt.

Convergence-

Controlling

-10

Saddle Pt.

10

Minimum

Error

Smallest

Term

-15

10

0 1 2 3 4 5

t

Figure 13. The integrand of the example of Balian, Parisi and Voros: exp(’φ). The

dominant saddle point is at t = 0. The secondary peak (saddle point) at t = 2

controls the asymptotic form of the coe¬cients of the asymptotic series; because

of it, the series for dt/dw converges only for | w |¤| w(t = 2) | . However, the

contribution of the more distant saddle point at t = 3 dominates the error. The

terms of the power series in 1/z reach a minimum at roughly exp(’zφ(t = 2)), but

the error in the optimally“truncated series is exponentially large compared to this

minimum term, being roughly exp(’zφ(t = 3))

where S( ) is the Stieltjes function. The asymptotic expansion for this

function is the same as for the Stieljes function; because exp(’(1/2)/ )

and all its derivatives vanish as ’ ∞, this function makes no contri-

bution to the divergent power series of f ( ). It follows that if we manip-

ulate the power series in the usual way, we arrive at a superasymptot-

ic approximation which, from the size of the smallest term, has an