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gral is, with a change in integration variable, proportional to Dingle™s
singular terminant.
The integral is not completely speci¬ed until one makes a choice
about how to deal with the pole on the path of the integration. Since
we know that for the Airy function, Stokes™ multiplier must increase
from 0 for real, positive z to 1 for real, negative z, the proper choice is
to indent the path of integration above the pole.
The integral is also not fully determined until the optimal truncation
Nopt has been identi¬ed. However, the coe¬cients asymptotic series for
E’ , which is the multiplier of the exponential which is dominant near
the Stokes line arg(z) = 2π/3, are asympotically factorials, just the
same as for the Stieltjes function (Eq. (81)). This implies that our
earlier analysis for S( ) applies here, too, to suggest

Nopt = |F | (86)

When F is real and positive, that is, when z is on the Stokes line,
the factor

χ ≡ exp(F (1 ’ t)) tNopt = exp {F (1 ’ t) + Nopt log(t)} (87)

has its maximum at t = 1, which coincides with the singularity of the
factor 1/(1’t) which is rest of integrand for the Stokes multiplier. This
coincidence of the saddle point with the pole requires only a slight
modi¬cation of standard descent to approximate M near the Stokes
line. Wong[319](pg. 356-360) gives a good discussion, attributing the
original analysis to van der Waerden [296].




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50 John P. Boyd

The key idea is to expand the factor χ as a power series about t = 1,
rather than the saddle point, which is slightly shifted away from t = 1
when (F ) = 0. Let T ≡ t ’ 1 and F ≡ Fr + iFim . Furthermore,
since the integral is strongly peaked about T = 0, the lower limit
of integration has been extended from T = ’1 to ’∞. The Stokes
multiplier is approximately

i 12 1
M=’ exp ’i Fim T ’ ’ Fr T 2 dT (88)
2π 2 T
’∞

where terms of O(T 3 ) in the exponential have been neglected.
This approximation is just the Fourier transform of a Gaussian func-
2
tion (exp(’(1/2)Fr T ), divided by iT . The identity

’i 1 1 1 x
exp(’ikx) exp(’a2 k 2 )
lim dk = + erf
δ’0 2π (k + i δ) 2 2 2a
’∞
(89)
shows that the Stokes multiplier is
1 1 Fim
+ erf √
M= (90)
2 2 2Fr
Fig. 11 shows that this approximation is very accurate.
The error function does not cover all cases; Chapman[95] has shown
that other smoothing functions are needed in some circumstances. How-
ever, the complementary error function does remove the “Victorian
discontinuity” of Stokes for a remarkably wide class of functions.


14. Matched Asymptotic Expansions in the Complex Plane:
The PKKS Method

In “above-the-barrier” quantum scattering, there are no turning points
where the coe¬cient of the undi¬erentiated term in the Schr¨dinger
o
equation is zero except at complex values of the spatial coordinate.
When there are real-valued turning points, it was discovered in the
1920s that the scattering ” including the exponentially small transmis-
sion through the barrier “ can be computed by means of the so-called
turning point connection formulas. (The transmission coe¬cient can be
calculated without heroics because the exponentially small transmitted
wave is the whole solution on the far side of the barrier, isolating it
from terms proportional to powers of as noted earlier.) Later, it was
shown that the connection formulas are really just a special case of the
method of matched asymptotic expansions [229, 20, 21]. The solution
in the neighborhood of the turning point can be expressed (to lowest




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51
Exponential Asymptotics

Stokes multiplier, approximated by terminant
0.5
0.99
0.7
14 0.9



0.01
12 0.2


0.3 0.4

10
|z|




0.6

8

0.1
6

0.8
4

1.9 2 2.1 2.2 2.3
θ/π

Figure 11. Solid: contours of the integral approximating the Stokes multiplier for
the Airy function. Dashed: contours of the error function approximation to this
integral. The solid and dashed contours are almost indistinguishable, which is a
graphical demonstration that the steepest descent approximation to the integral is
very accurate.


order) in terms of the Airy function Ai. This is matched to standard
WKB approximations which describe the solution everywhere else.
For “above-the-barrier” scattering, however, what is one to do?
Pokrovskii and Khalatnikov [262] had a ¬‚ash of insight: actually, there
are turning points, but only for complex x. In the vicinity of these
o¬-the-real-axis turning points, the re¬‚ected wave is not small, so the
usual connection formulas apply with only minor modi¬cations. The
amplitude of the re¬‚ected wave decays exponentially as (x) ’ 0 so
that, on the real x-axis, the re¬‚ection coe¬cient is exponentially small
in 1/ , the inverse width of the barrier.2
Kruskal and Segur [171, 172, 278] showed that matching expan-
sions at o¬-the-real-axis critical points was a powerful method for non-
linear problems, too. Their ¬rst application resolved a forty year old
conundrum in the formation of multi-branched ¬ngers (“dendrites”)
on a solid-liquid interface. The unique length scale observed in the
2
Pokrovskii relates an amusing story: when he presented his work to the Nobel
laureate, Lev Landau, the great man thought he and Khalatnikov were crazy! He
eventually changed his mind.[261]




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52 John P. Boyd

laboratory is imposed by surface tension. However, the scale-selecting
e¬ect lies “beyond all orders” in a power series expansion in the surface
tension parameter. Their method, which we shall henceforth call the
“PKKS” [Pokrovskii-Khalatnikov-Kruskal-Segur] scheme for short, has
been widely used for weakly nonlocal solitary waves (Table III).
To illustrate the PKKS method, we shall apply it to the linear prob-
lem:
uxx + u = f ( x) (91)
where f (x) will be restricted to functions that (i) decay exponential-
ly as | x |’ ∞ on and near the real axis and (ii) have a complex
conjugate pair of double poles at x = ±i as the singularities nearest
the real axis and (iii) are symmetric with respect to x = 0, that is,
f (x) = f (’x). This seems like a rather special and restrictive prob-
lem. However, as Dingle observed long ago, every simple example is a
master-key to an entire class of problems, as we shall show. This linear
problem is identical to that solved earlier, Eq.(26), except for the sign
of the undi¬erentiated term in u.
We shall impose the boundary condition that

u ∼ ± sin(|x|) as |x| ’ ∞ (92)

for some constant ± which will be determined as part of the solution.
This excludes the homogeneous solutions sin(x) and cos(x) so as to
yield a unique solution. (Note the absolute value bars inside the argu-
ment of the sine function in the boundary condition.)
The PKKS method has the following steps:

1. Identify the singularities or critical points which are nearest the
real x-axis

2. De¬ne an “inner” problem, that is, a perturbative scheme which
is valid in the neighborhood of one of these critical points, using a
complex coordinate y whose origin is at the critical point.

3. Asymptotically solve the “inner” problem as | y |’ ∞, that is,
compute the “outer limit of the inner solution”.

4. Sum the divergent outer limit of the inner problem by Borel sum-
mation or otherwise determine the connection formula, that is, the
magnitude and phase of the discontinuity along the Stokes line
radiating from the critical point to the real x-axis

5. Match the outer limit of the inner solution to the inner limit of the
outer expansion




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53
Exponential Asymptotics




Real y-axis
arg(y)=0
Inner




Im(t)
Re(t)
Matching
arg(y)=
- π/2
arg(y)=
Outer arg(y)=
-π -(3/4) π
Real x-axis


arg(y)=
-(1/4) π
Im(t)
Im(t)




Re(t)
Re(t)

arg(y)=
-(3/4) π

Figure 12. (a) [Upper left corner] Schematic of complex y-plane where y is the
shifted coordinate. (The real axis in the original coordinate x is the arrow at the
bottom.) The location of the double pole (at y = 0) is the large solid dot at top.
The “matching” region, shaped like a half annulus, is where both the inner and
outer solutions are valid, allowing them to be matched. (b) [Upper right corner] The
complex t-plane where t is the integration variable for the Borel-logarithm function,
Bo(y). The four large black discs show the location of the logarithmic singularity
of the integrand for four di¬erent values of arg(y). The branch cut (cross-hatched
lines) goes to i ∞ for all cases. As arg(y) increases, the location of the branch cut
rotates clockwise. For arg(y) < ’π/2, the branch cut crosses the real t-axis as shown
in the lower right half diagram. (c) [Bottom half of the ¬gure]. Both left and right
panels illustrate the path of integration in the complex t-plane (heavy, patterned
curves) and the branch cuts for the logarithm of the integrand (cross-hatched lines).
The left diagram shows the situation when arg(y) = ’π/4, or any other point such
that the branch point is in the upper half of the t-plane: the branch cut does not
cross the real axis. When arg(y) < ’π/2 [right, bottom diagram], the integration
path must be deformed below the real t-axis to avoid crossing the branch cut. The
integration around the branch cut adds an additional contribution.




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