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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