Stokes phenomenon and matched asymptotic expansions

TLDR: The use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon and highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymPTotic expansions.

Abstract: This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions.

Full text

SIAM
J.
APPL.
MATH.
Vol.
55,
No.
6,
pp.
1469-1483,
December
1995
@1995
Society
for
Industrial
and
Applied
Mathematics
001
STOKES
PHENOMENON
AND
MATCHED
ASYMPTOTIC
EXPANSIONS*
A.
B.
OLDE
DAALHUIS-,
S.
J.
CHAPMAN,-,
J.
R.
KINGS,
J.
R.
OCKENDON,
AND
R.
H.
TEW$
Abstract.
This
paper
describes
the
use
of
matched
asymptotic
expansions
to
illuminate
the
description
of
functions
exhibiting
Stokes
phenomenon.
In
particular
the
approach
highlights
the
way
in
which
the
local
structure
and
the
possibility
of
finding
Stokes
multipliers
explicitly
depend
on
the
behaviour
of
the
coefficients
of
the
relevant
asymptotic
expansions.
Key
words.
Stokes’
phenomenon,
matched
asymptotic
expansions,
Airy
function,
error
func-
tion
AMS
subject
classifications.
41A60,
33C10,
33B20
1.
Introduction.
The
role
of
Stokes
phenomenon
in
describing
the
asymptotic
behaviour
of
an
analytic
function
as
its
argument
tends
to
an
isolated
singularity
has
been
studied
intensively
in
recent
years
(Berry
[1],
Berry
and
Howls
[3],
McLeod
[4],
Meyer
[5],
Olde
Daalhuis
and
O1ver
[8],
O1ver
[10],
Paris
[12],
Paris
and
Wood
[13]).
As
originally
discussed
by
Stokes,
the
basic
picture
is
that
an
asymptotic
expansion
of
the
function
that
is
uniform
in
phz
can
be
constructed
only
if
an
exponentially
small
correction
(in
terms
of
distance
from
the
singularity)
is
made
as
certain
directions
are
traversed
in
the
Argand
diagram.
These
directions
are
called
Stokes
lines,
and
when
the
function
under
consideration
is
a
complementary
function
of
a
certain
class
of
linear
holomorphic
second-order
differential
equations,
they
are
characterised
as
lines
where
one
complementary
function
is
maximally
dominant
over
another.
The
Stokes
lines
are
separated
by
other
directions,
called
anti-Stokes
lines,
which
are
practically
important
because,
across
them,
a
complementary
function
switches
from
being
dominant
to
subdominant;
however,
there
is
no
nonuniformity
in
their
vicinity,
and
all
the
action
takes
place
near
the
Stokes
lines.
What
has
emerged
recently
is
the
detailed
structure
of
the
behaviour
in
the
vicin-
ity
of
Stokes
lines,
at
least
for
a
class
of
functions
whose
asymptotic
expansions
diverge
in
a
certain
way.
Thus,
instead
of
the
traditional
asymptotic
representation
in
terms
of
divergent
expansions
in
different
sectors,
with
discontinuous
coefficients
that
are
related
by
the
Stokes
connection
formulae,
the
smoothness
inherent
in
the
analyticity
can
be
restored,
and
this
can
be
done
universally
in
terms
of
error
functions
[1].
The
purpose
of
this
paper
is
to
interpret
this
state
of
affairs
in
terms
of
theory
of
matched
expansions
(MAEs)
(van
Dyke
[14]),
not
just
with
the
aim
of
simplifying
the
representation
of
the
above-mentioned
new
developments
but
also
to
pave
the
way
for
these
developments
to
be
exploited
in
problems
other
than
the
linear
ordinary
differential
equations
(ODEs)
that
have
been
considered
hitherto.
To
fix
ideas,
we
will
begin
by
recalling
Stokes
phenomenon
for
some
linear
ODEs
where
there
are
explicit
integral
representations.
These
examples
will
then
be
used
*Received
by
the
editors
January
19,
1994;
accepted
for
publication
(in
revised
form)
October
31,
1994.
Mathematical
Institute,
University
of
Oxford,
24-29
St.
Giles,
Oxford
OX1
3LB,
United
Kingdom.
:Department
of
Theoretical
Mechanics,
University
of
Nottingham,
Nottingham
NG7
2RD,
United
Kingdom.
1469

1470
OLDE
DAALHUIS
CHAPMAN
KING
OCKENDON
TEW
to
motivate
the
construction
of
the
MAE
framework
with
which
we
will
be
working.
It
will
first
enable
us
to
study
the
relevance
of
the
phenomenon
for
a
class
of
linear
homogeneous
ODEs
in
order
to
explain
how
to
decide
when
connection
formulae
can
be
calculated
explicitly
rather
than
by
numerical
computation.
Then
we
use
the
MAE
format
to
present
new
results
concerning
the
asymptotic
behaviour
of
inhomogeneous
ODEs.
Finally,
in
the
conclusion,
we
will
be
able
to
make
some
conjectures
about
the
applicability
of
our
framework
to
other
ODEs
and
partial
differential
equations
(PDEs).
A
crucial
aspect
of
our
approach
is
the
reinterpretation
of
Stokes
phenomenon
in
terms
other
than
that
of
maximal
dominance
of
complementary
functions
of
ODEs.
One
interpretation
that
emerges
naturally
from
Berry
[I]
is
that
of
the
rapidity
with
which
the
difference
between
the
solution
of
an
ODE
and
its
optimally
truncated
asymptotic
expansion
varies
as
ph
varies.
This
is
tied
in
with
the
description
of
Stokes
phenomenon
not
as
a
change
in
the
coecient
of
a
complementary
function
but
rather
as
a
change
in
the
remainder
of
the
asymptotic
expansion
of
the
dominant
complementary
function
in
the
region
where
this
remainder
is
comparable
with
the
subdominant
complementary
function.
This
remainder
or
error
will
have
relatively
smooth
variation
away
from
a
Stokes
line,
and
the
change
traditionally
ascribed
to
the
Stokes
multiplier
is
a
aqa
of
the
rapid
variation
in
this
error
rather
than
its
cause.
However,
yet
another
characterisation
has
been
proposed
by
Wright
[15]
in
terms
of
the
equality
of
the
phase
of
certain
solutions
of
linear
PDEs
when
inter-
preted
as
waves,
and
this
is
the
one
that
will
emerge
most
naturally
as
a
result
of
our
investigation.
.
Examp|e
The
complementary
error
function.
The
complementary
ror
function
s
a
well-known
function
defined
by
(2.1)
erfc(z)
e
dr,
z
C.
It
is
an
entire
function
with
the
following
asymptotic
behaviour:
(2.2a)
erfc(z)
e-
(2s)l
3
z
0(-1)
sl(4z2)
]Ph(z)l
<
w’
e
(2s)
3
rfc( )
+
<
s0
as
[z[
--
cx.
Notice
that
both
asymptotic
expansions
are
valid
in
the
sector
7r
<
phz
<
}7r.
So
the
asymptotic
behaviour
of
erfc(z)
is
the
infinite
expansion
of
(2.2a),
plus
a
constant.
In
the
sector
-r
<
phz
<
r,
where
exp(-z2)/zv/-
is
subdominant,
this
constant
is
0.
In
the
sector
r
<
phz
<
7r,
where,
again,
exp(-z2)/Zv/-
is
sub-
dominant,
this
constant
is
2.
And
in
the
sector
r
<
phz
<
r,
where
exp(-z)/zx/
is
dominant,
this
constant
changes
from
0
to
2
(see
Fig.
2.1).
We
want
to
obtain
the
change
of
the
asymptotic
behaviour
directly
from
the
differential
equation
d
2
d
w
+
o,
without
use
of
(2.1).
Both
erfc(z)
and
the
constant
function
are
solutions
of
this
differential
equation.
To
obtain
more
information
on
the
change
in
the
sector
1/47r
<

STOKES
PHENOMENON
AND
MATCHED
ASYMPTOTIC
EXPANSIONS
1471
exp
FIG.
2.1.
The
asymptotic
behaviour
of
erfc(z)
for
0
<_
phz
<_
r.
phz
<
}r,
we
truncate
(2.2a)
after
N
terms:
(2.4)
erfc(z)
e-
(-1)
s!(4z2)-----
+
RN(Z).
ZV/
s=0
The
remainder
RN(Z)
is
a
solution
of
d2
e-z
(--1)N(2N)!
dz
2
RN(Z)
+
2z
RN(Z)
zx/
(N-
1)!4N-lz
2N"
To
show
where
the
significant
changes
take
place,
we
introduce
polar
coordinates
(2.6)
d
ie
-
d
d2
ie
-
d
e
-
d
z
re
i,
O
<
O
<
Tr,
dz
r
dO’
dz
2
r
2
dO
r
2
dO
2’
where
we
have
deliberately
written
z
in
terms
of
the
"fast"
variable
0
rather
than
the
"slow"
variable
r.
Equation
(2.5)
in
terms
of
polar
coordinates
is
(2.7)
r2
dO
2
N(Z)’}-i
T2
2
-RN(Z)
exp[-r2e
2
+
UTvi-
(2N
+
1)iO](2N)!
x/(N-
1)!4N-rU+
The
magnitude
of
the
right-hand
side
of
(2.7),
as
a
function
of
N,
is
minimal
for
N
r
.
Therefore,
we
take
N
r
2
+
a,
where
a
is
bounded
as
r
c.
With
this
N
the
right-hand
side
of
(2.7)
reads
(2.8)
exp[-r2(e
2i
+
i(20
7r))
+
aTri
(2a
+
1)Oi](2r
+
2a)!
V(r
2
+
a
1)!4r+a-lr
2r+2+1
8r
e-(-)-exp[-r(e
+
1
+
i(20-
))1
as
r
-
c.
The
dominant
factor
in
the
right-hand
side
of
(2.8)
is
lexp[-r2(e
2i
d-
1
+
i(20-
r))]l,
and
it
is
maximal
at
0
1/27r,
where
it
is
O(1),
as
r
--
xz.
The
value
0
1/2r
has
two
other
special
properties.
First,
the
phase
of
the
second
exponential
is
both
zero
and
stationary
at
this
point
(so
that
the
right-hand
side
ceases
to
be
oscillatory
as
r
-
cxz
in
the
vicinity
of
the
Stokes
line).
Second,
and
most
importantly

1472
OLDE
DAALHUIS
CHAPMAN,
KING,
OCKENDON,
TEW
for
the
present
point
of
view,
the
right-hand
side
of
(2.8)
is
independent
of
c
when
0
Tr.
This
latter
property
allows
us
to
use
matched
asymptotic
expansions
to
solve
for
RN
in
the
neighbourhood
of
the
Stokes
line.
We
write
ro
1
(2.9)
r=--
0=
7r+500,
where
e
and
5
are
the
new
small
parameters.
With
the
substitution
of
(2.9)
into
(2.7),
we
obtain
as
5,
e
--
0.
From
the
final
exponential
we
only
obtain
an
interesting
result
when
5
e.
Then
the
dominant
terms
are
the
e-1
terms,
and
we
obtain
(2.11)
dOo
t:tN(Z)
4exp(_2r00),
VzTr
with
the
solution
ro
1/2r+eOo)
(2.12)
IN(Z)
A
+
erf(v/roOo),
z
--e’(
--
O,
where
erf(z)
is
the
error
function.
Matching
as
00
--,
-oc,
the
remainder
Rw(z)
is
exponentially
small.
Thence,
A
1.
Thus
all
the
change
in
the
constant
term
takes
place
in
the
neighbourhood
of
the
Stokes
line
phz
1/27r,
and
the
change
reads
(2.13)
RN(Z)
1
+
erf(Vr000),
in
agreement
with
[1].
Figure
2.2
shows
the
appearance
of
the
constant
term
in
the
asymptotic
behaviour
of
erfc(z)
in
the
sector
0
_<
phz
_<
2r.
Notice
that
the
constant
5
term
appears
at
the
Stokes
line
phz
Tr,
it
is
dominant
in
the
sector
Tr
<
phz
<
Tr,
and
it
disappears
at
the
Stokes
line
phz
Tr.

STOKES
PHENOMENON
AND
MATCHED
ASYMPTOTIC
EXPANSIONS
1473
3.
The
Stokes
phenomenon
for
solutions
of
a
class
of
ODEs.
The
general
homogeneous
linear
differential
equation
of
the
second
order
is
given
by
(3.1)
d
2
d
+
+
o.
We
suppose
that
the
point
at
infinity
is
an
irregular
singularity
of
rank
1.
The
asymp-
totic
theory
of
solutions
of
(3.1)
in
these
circumstances
is
well
known
and
will
be
found,
for
example,
in
Olver
[9,
Chap.
7,
1-2].
Without
loss
of
generality
we
may
assume
that
f(z)
and
g(z)
can
be
expanded
in
the
power
series
(3.2)
f
(z)
1-
p
+
2-
g(z)
E
g8
Z
Z
-
8"-’2
which
converge
for
Izl
>_
p.
The
two
unique
solutions
of
(3.1)
have
the
following
asymptotic
behaviour"
as
3
(3.3a)
Vl(Z)
e-zz"
E
-Z-’
Iph(z)[
<
Tr
,
8--0
3
(3.35)
v2(z)
1,
Iph(z)
7r
_<
Tr-
.
Here
and
elsewhere
in
the
paper
denotes
an
arbitrary
small
positive
parameter.
We
choose
a0
1,
and
the
other
coefficients
are
determined
by
s--1
s--2
(3.4
-sa8
s(s
1
t)as-1
-4-
E
(gs+l-m
fs+l--m)am
E
(m
#)L-,a,,
s
>_
1.
A
direct
consequence
of
(3.2)
is
that
Vl
(Ze
-27ri)
is
also
a
solution
of
(3.1),
and
note
that
vl(z)
and
e2"Vl(ze
-2i)
are
dominant
solutions
in
the
sector
1/27r
+
<
3
phz
<
Tr-
and
have
exactly
the
same
asymptotic
expansion
there.
Thus
there
is
a
Stokes
multiplier
C
such
that
Vl
(Z)
e27riPvl
(ZC
-2r{
-}-
CV2(Z).
With
this
connection
formula
we
obtain
the
following
asymptotic
behaviour
for
vl
(z)’
1
1
(3.6a)
Vl(Z)
e-zz
-Tr
<
phz
<
r,
subdominant
2
1
3
(3.65)
v
(z)
e-zz,
r
<
phz
<
7r,
dominant,
3
5
(3.6c)
v(z)
C,
r
<
phz
<
Tr,
dominant.
3
Again,
somewhere
in
the
sector
gr
<
phz
<
-,
new
exponentially
small
terms
appear.
To
obtain
more
information
concerning
the
change
in
this
sector,
we
truncate
(3.3a)
after
N
terms
N-1
(3.7)
vl
(z)
e-zz"
E
a-2-
+
RN(Z).
Z
8=0