ˆ

with B(z) = 0 An z ’n .

r

ˆ

2. Let A = Λ + N , with a diagonalizable matrix Λ and nilpotent N

commuting with Λ (observe that such a decomposition always exists).

If B commutes with A, show that B commutes with both Λ and N .

3. Let N be a nilpotent matrix. Show that T (z) = exp[m’1 z m N ],

m ∈ N, is a meromorphic transformation. In case N commutes with

all An , show that

B(z) = T ’1 (z) A(z) T (z) ’ z T (z) = A(z) ’ z m N.

ˆ ˆ ˆ

4. Given an elementary formal system (3.3), show that there exists a

ˆ

formal meromorphic transformation T (z) so that the transformed

system has a coe¬cient matrix B(z) of the form

B(z) = diag [b1 (z) Is1 + N1 , . . . , bµ (z) Isµ + Nµ ],

with distinct polynomials bm (z) of degree at most r satisfying 0 ¤

Re bm (0) < 1, 1 ¤ m ¤ µ, and nilpotent constant matrices Nm ,

whose blocks are increasing in size.

8.5 De¬nition of Highest-Level Normal Solutions 137

5. For B(z) as above, compute a fundamental solution of the system

z x = B(z) x.

˜ ˜

6. Let N1 , N2 be nilpotent matrices, not necessarily of the same size. Let

ˆ ˆ ˆ

T (z) be a formal meromophic series satisfying z T (z) = N1 T (z) ’

ˆ ˆ

T (z) N2 . Show that then T (z) is constant.

8.5 De¬nition of Highest-Level Normal Solutions

After this digression into the classical subject of FFS, we now continue with

our investigation of HLFFS. For this purpose, we consider a system (3.1)

ˆ

together with a ¬xed HLFFS F (z), and we shall use the same notation

as on p. 55 for its data pairs (qj (z), sj ), 1 ¤ j ¤ µ, and the coe¬cient

matrix B(z) of the transformed system. From the data pairs we then can

ˆ

explicitly compute the singular directions for F (z) as the solutions of (8.3),

with k = r ’ p/q, and we shall from now on denote them by dj , labeled so

that

dj’1 < dj , j ∈ Z, d’1 < 0 ¤ d0 . (8.8)

Interchanging j and m in (8.3), we conclude that dj is singular if and only

if dj ± π/k is so, too. The data pairs are closed with respect to continu-

(p)

ation, so in particular, the set {»n , 1 ¤ n ¤ µ} is closed with respect

to multiplication by exp[2πi/q]. Hence, we also see that dj ± 2π/(qr ’ p)

is again a singular direction, for every integer j. Together, this shows the

following:

• Let µ denote the greatest common divisor of q and 2; in other words,

µ = 2 whenever q is even, and µ = 1 for odd q. Then for every singular

direction dj , the directions dj ± µπ/(qr ’ p) are singular as well. In

other words, every half-open interval of length µπ/(qr ’ p) contains

the same number of singular directions, say, j1 . Set j0 = 2(qr’p) j1 /µ;

hence j0 = (qr ’ p) j1 when q is even, resp. j0 = 2(qr ’ p) j1 for odd

q. Then j0 is the number of singular directions in every half-open

interval of length 2 π, or equivalently dj±j1 = dj ± 2 π for every j ∈ Z.

Let Sj = { z : |z| > ρ, dj’1 ’ π/(2k) < arg z < dj + π/(2k)}. These are

sectors near in¬nity whose boundary rays are usually referred to as Stokes™

directions 3 because of their signi¬cance in the Stokes phenomenon (see the

following chapter). We choose to denote these directions as „j = dj +π/(2k),

j ∈ Z.

3 In the literature one sometimes ¬nds the term anti-Stokes™ directions for what we

call the singular directions dj .

138 8. Solutions of Highest Level

ˆ

For each d in the interval (dj’1 , dj ), the series F (z) is going to be k-

summable (with k = r ’ p/q), and according to Lemma 10 (p. 101) the

ˆ

sum of F (z) is independent of d, but will in general depend on j, hence we

choose to denote this sum by Fj (z).

Theorem 44 The matrix functions Fj (z) de¬ned above are holomorphic

for |z| > ρ on the Riemann surface of the logarithm, and there we have

zFj (z) = A(z) Fj (z) ’ Fj (z) B(z), j ∈ Z. (8.9)

Moreover, they satisfy Fj (z) ∼1/k F (z) for z ’ ∞ in the sector Sj .

ˆ

=

Proof: It follows from the general theory of k-summability that Fj (z) is

holomorphic in some sectorial region G at in¬nity with opening dj ’ dj’1 +

π/k, and bisecting direction (dj + dj’1 )/2. Moreover, Fj (z) ∼1/k F (z)

ˆ

=

ˆ

there. Since F (z) formally satis¬es

ˆ ˆ ˆ

z F (z) = A(z) F (z) ’ F (z) B(z),

we conclude from the algebraic properties of k-summability that (8.9) holds

in G. Since (8.9) is nothing but a system of linear ODE for the entries

of Fj (z), we conclude from the general theory in Chapter 1 that Fj (z)

can be holomorphically continued along arbitrary paths within the region

{|z| > ρ}, hence is holomorphic on the corresponding part of the Riemann

surface of the logarithm. Every closed subsector of Sj with su¬ciently large

radius is contained in the sectorial region G, so the asymptotic holds in Sj .

2

In what follows, we shall use the term highest-level normal solutions for

the matrices Fj (z) de¬ned above, and we shall use the abbreviation HLNS.

We shall study these HLNS in greater detail in the following chapter.

Exercises: In the following exercises we consider two linear systems

z xj = Aj (z) xj for |z| > ρ, of arbitrary Poincar´ ranks at in¬nity and

e

of dimension νj , with ν1 not necessarily equal to ν2 .

1. Verify that z X = A1 (z) X ’ X A2 (z), |z| > ρ, may be rewritten as

a linear system of meromorphic ODE in dimension ν1 · ν2 .

2. Let X(z) be a solution of the linear system of ODE in the previous

exercise, say, near some point z0 , |z0 | > ρ. Show that then X(z) can

be holomorphically continued along arbitrary paths within R(∞, ρ).

In case ν1 = ν2 , show that det X(z0 ) = 0 implies det X(z) ≡ 0.

3. For the HLNS, de¬ned above, show that det Fj (z) = 0 for arbitrary

values z with |z| > ρ.

9

Stokes™ Phenomenon

The fact that a solution of an ODE, near an irregular singularity, in dif-

ferent sectors of the complex plane in general shows di¬erent asymptotic

behavior was observed and studied by Stokes [258] and is, therefore, named

Stokes™ phenomenon. An equivalent way of describing this phenomenon in

ˆ

the language of summability is as follows: Consider an HLFFS F (z), as

ˆ

de¬ned in Section 3.5, of some system (3.1) (p. 37). We have shown F (z)

to be k-summable, for a unique k > 0, in all but some singular directions,

thus de¬ning the HLNS Fj (z). In corresponding sectors, all Fj (z) have the

same asymptotic expansion, despite of the fact that in general they do de-

pend upon j: According to Proposition 13 (p. 105), the matrices Fj (z) are

ˆ

independent of j if and only if the HLFFS F (z) converges. Hence in other

words, Stokes™ phenomenon is caused by the divergence of formal solutions!

In this chapter, we shall investigate how two consecutive Fj (z) are in-

terrelated, and in this way analyze Stokes™ phenomenon of highest level.

Classically, this part of the theory was based upon the choice of a FFS;

see, e.g., [34, 35] or [143]. This has the serious disadvantage of mixing the

phenomena arising on di¬erent levels, as shall become clearer later on. So

this is why we shall work with HLFFS instead.

In principle, Stokes™ phenomenon is described by ¬nitely many constant

matrices, usually named Stokes™ multipliers, or lateral connection matrices.

We shall explicitly describe their structure, and discuss in which sense they

can be computed in terms of the HLFFS used in their de¬nition. In some

cases of dimension ν = 2, we shall obtain explicit values for the multipliers

in terms of the data of the system. We then present a result, ¬rst obtained

by Birkho¬ [55, 57], on the freedom of the Stokes multipliers. For another

140 9. Stokes™ Phenomenon

proof of this fact using a somewhat di¬erent approach, see the monograph

of Sibuya [251].

Many authors have been concerned with the theoretical investigation

and/or numerical computation of Stokes multipliers, e.g., Turrittin [267,

272], Heading [115, 116], Okubo [205], Hsieh and Sibuya [130], Sibuya [247“

250, 252], Kohno [154, 155, 160], Smilyansky [256], Braaksma [66“69], Goll-

witzer and Sibuya [109], McHugh [189], Jurkat, Lutz, and Peyerimho¬ [146“

149], Emamzadeh [97“99], Hsieh [129], Sch¨fke [237, 240], Hinton [121],

a

Balser, Jurkat, and Lutz [33, 37“39, 41], Gurarij and Matsaev [111], Ramis

[228, 227], Balser [11“13, 16, 20], Slavyanov [255], Schlosser-Haupt and

Wyrwich [243], Duval and Mitschi [93], Babbitt and Varadarajan [4], Tovbis

[264, 265], Lin and Sibuya [168], Immink [135“137], Tabara [259], Kostov

[162], Naegele and Thomann [197], Sibuya and Tabara [254], and Olde Dal-

huis and Olver [211]. Also compare Section 12.5 on the so-called central

connection problem.

9.1 Highest-Level Stokes™ Multipliers

Throughout what follows, we consider a ¬xed system (3.1) (p. 37), having

ˆ

an essentially irregular singularity at in¬nity. Let F (z) be an HLFFS of

(3.1), together with the corresponding HLNS Fj (z). Recall that each Fj (z)

satis¬es (8.9) (p. 138), with B(z) diagonally blocked of type (s1 , . . . , sµ ).

We arbitrarily choose a fundamental solution Y (z) = diag [Y1 (z), . . . , Yµ (z)]

of z y = B(z) y. It follows from (8.9) that Xj (z) = Fj (z) Y (z) is a solu-

tion of (3.1). Since det Xj (z) = 0 follows from Exercise 3 on p. 138, every

Xj (z) is in fact a fundamental solution of (3.1). From the exercises be-

low, we then conclude that every solution of (8.9) can be represented as

Xj (z) C Y ’1 (z), for a suitable constant matrix C. So in particular, there

exist unique matrices Vj such that

Fj’1 (z) ’ Fj (z) = Fj (z) Y (z) Vj Y ’1 (z), |z| > ρ, j ∈ Z. (9.1)

This formula is equivalent to Xj’1 (z) = Xj (z) (I + Vj ), for j ∈ Z, and we

shall refer to the matrices Vj as the Stokes multipliers of highest level of the

ˆ

system (3.1). Note that Vj depends upon both F (z) and Y (z). Therefore, we

ˆ

here shall always consider pairs (F (z), Y (z)), and for simplicity of notation

we also use the term HLFFS to refer to such a pair. From (8.9) we conclude

that the left-hand side of (9.1) is asymptotically zero of Gevrey order 1/k

in Sj © Sj’1 , and we shall use this to investigate the structure of the Stokes

multipliers Vj :

With qn (z) as in the de¬nition of the data pairs of HLFFS, we set

Gn (z) = e’qn (z) Yn (z). From the special form of B(z), and using Exer-

cise 4 on p. 52 together with Exercise 6 on p. 7, we conclude that each

Gn (z) and its inverse are of exponential growth strictly less than k. Hence,

9.1 Highest-Level Stokes™ Multipliers 141

the behavior of Yn (z) near in¬nity is in principle determined by that of

(j)

qn (z). Blocking Vj = [Vnm ] in the block structure of Y (z), the matrix

Y (z) Vj Y ’1 (z) on the right-hand side of (9.1) is likewise blocked, and the

behavior of a block, as z ’ ∞, depends upon whether or not we have

exp[qn (z) ’ qm (z)] = exp[(»(p) ’ »(p) )z k /k] ∼1/k ˆ

0 (9.2)

=

n m

for z ∈ Sj’1 ∪ Sj . For a pair (n, m), this holds if and only if ’kdj’1 =

(p) (p)

arg(»m ’ »n ) mod 2π. In other words, this characterizes pairs (n, m) for

which the left-hand side of (9.2) has maximal rate of descend along the ray

arg z = dj’1 . The set of all such pairs will be denoted by Suppj . Note that

we always have (n, n) ∈ Suppj , and Suppj is never empty, owing to the

de¬nition of dj’1 . The next theorem says that the set of these pairs is “the

support” of Vj :

Theorem 45 For a system (3.1) with an essentially irregular singularity

ˆ

at in¬nity, let (F (z), Y (z)) be a given HLFFS. Then for the corresponding

(j)

Stokes multipliers of highest level, (n, m) ∈ Suppj implies Vnm = 0. Hence,

in particular all the diagonal blocks of Vj vanish.

Proof: From (9.1) we conclude Y (z) Vj Y ’1 (z) ∼1/k ˆ in Sj’1 ∪ Sj . Using

0

=

(j) ’qm (z) (p)

Exercise 7 on p. 74, this is equivalent to eqn (z) Vnm e = exp[(»n ’

»m ) z k ] Vnm ∼1/k ˆ in Sj © Sj’1 , for every (n, m), since Gn (z) and G’1 (z)

(p) (j)

0

= m

both are of exponential growth strictly less than k. According to the de¬-

2

nition of Suppj , this completes the proof.

Exercises: In the ¬rst two exercises to follow, we consider two linear

systems zxj = Aj (z) xj for |z| > ρ, of arbitrary Poincar´ ranks at in¬nity

e

and of dimension νj , with ν1 not necessarily equal to ν2 .

1. Let Xj (z) be a fundamental solution of zxj = Aj (z) xj . For C ∈

’1

C ν1 —ν2 , show that X1 (z) C X2 (z) is a solution of

zX = A1 (z) X ’ X A2 (z), |z| > ρ. (9.3)

2. Let X(z) be any solution of (9.3). Verify that X(z) can be represented

’1

as X1 (z) C X2 (z) for a unique C ∈ C ν1 —ν2 .

3. For every pair (n, m), 1 ¤ n < m ¤ µ, show: Every half-open inter-

val of length π/k contains exactly one direction „ where, for every