0387986995Basic TheoryC

368 XI. ASYMPTOTIC EXPANSIONS

X1-9. Consider a formal power series

(p21, ... , p2",) with non-negative integers P1k and p2k. Denote by Fg(x, y z) the Jacobian matrix of ! with respect to W. Assume that power series (P) satisfies the conditions (1) f;(0,6,6) = 0, (2) Fy(0, 0, 0') is invertible, (3) fn, a, E E(s, A)" for

convergent power series in y and F. Show that there exists a unique power series

¢(x, z) 92 in (x, z) such that (i) drr,, E C[[x]]", (ii) b(0,0) = 0, and (iii)

lack>o

F(x, (x, z), z) = 0. Also, show that (a, E E(s, B)" for some B > 0 and the power

Hint. This is an implicit function theorem. Write f in the form f = fo + fiy + , where fo and fn,,n, are in E(s, A)", $ E M"(E(s, A)), and >' is the sum over all (pi i p) such that either jell = 0 and [g-,j = I or Jial j +

jp2j > 2. Here, M.(R) denotes the set of all n x it matrices with entries in a ring

R. Assumption (1) implies that fo(0) = 0. Assumption (2) implies that -t(0) is invertible. Therefore, I ' exists in M"(C[[x]],), and hence II4 'jI, A < +oo for

some A > 0, where III-' IIB,A = sup{jj4V' f Ij, A : f E E(s, A)n, Ij f [j,.A = 11. Since

IIfIIe,B 5 11A .,A for f E (C[[x]],)" if B > A, assume without any loss of generality

9P% ,P2 Il .w

XI-13. Prove the formula

00

M=0

which is given in Therorem XI-2-4.

Hint. Assume that j argx1 < 2k - b, where b is a small positive number. Then,

/r +00

+k J cmtm e-(t/x)ktk-ldt.

xk 0 m=N+1

XI-14. Let a be a positive number larger than 1. Also, let

SJ={x: a, <argx<bj, 0<jxj <p) (1=1,2,... v)

be a good covering of the sector S = {x : j argxj < 'r,0 < jxj < p}, where a1 =

, b = , and p is a positive number. Assume that v functions &1(x), 4(x),

satisfy the following conditions:

(i) Oj(x) is holomorphic in Sj and continuous on the closed sector

19, ={x:aj5argx<bj,O<lxj<p},

(ii)103(x)1 < Aexp[clxl-1] in S, for some positive numbers A and c,

(iii)10,+1(x) - Oj(x)j < Mo in S, fl S,t1 for some positive number M0,

(iv)1O1(x)j < M0 on the line segment argx = a1, 0 < lxj < p,

(v)< M0 on the line segment argx = b,,, 0 < lxj < p.

Show that there exists a positive number M such that

Hint. This is a generalization of the Phragmen-Lindelof Theorem (Lemma XI-3-5).

See ]Lin].

CHAPTER XII

ASYMPTOTIC SOLUTIONS IN A PARAMETER

In this chapter, we explain asymptotic solutions of a system of differential equa-

tions E°fy = f (x, y, f) as a -i 0. In §§XII-1, XII-2, and XII-3, existence of such asymptotic solutions in the sense of Poincar6 is proved in detill. In §XII-4, this result

is used to prove a block-diagonalization theorem of a linear system E° d = A(x, e)y.

In §XII-5, we explain similar results in the Gevrey asymptotics. In §XII-6, we explain how much we can simplify a linear system by means of a linear transformation with a coefficient matrix whose entries are convergent power series in the param- eter. This result is given in [Hs1] and similar to a theorem due to G. D. Birkhoff

[Bi] concerning singularity with respect to the independent variable x (cf. Theorem

XII-6-1 and (Si17, Chapter III]). The materials in §§XII-1- XII-5 are also found in

[Wasl], [Si3], [Si7], and [Si22].

XII-1. An existence theorem

In §§XII-1, XII-2, and XII-3, we consider a system of differential equations

where a is a positive integer and f J (x, V1, v2, ... , vn, f) are holomorphic with respect to complex variables (x, v1, v2, ... , vn, E) in a domain

(XII.1.2) Ix[ < do, 0< IEI < Po, I arg EI < ao, Iv. I < 'ro (j = 1, 2, ... , n),

6o, po, ao, and 'Yo being positive constants. Set

where i I E Cn with the entries (V1, V2, ... , vn).

We look at (XII.1.1) under the following three assumptions.

Assumption I. Each function f j (x, v, c) (j = 1, 2,... , n) has an asymptotic ex- pansion

372

Observation XII-1-1. Under Assumption I, fjo(x, e), a jh(x, e), and f jp(x, e) ad- mit asymptotic expansions

as e - 0 in sector (XII.1.5) with coefficients holomorphic in the domain

(XII.1.10) 1x(< 60-

Let A(x, e) be the n x n matrix whose (j, k)-entry is a.,&, e), respectively (i.e.,

A(x, e) = (ajk(x, e)). Then, A(x, c) admits an asymptotic expansion

as a -+ 0 in sector (XII.1.5), where the entries of coefficient matrices

(ajk,,(x)) are holomorphic in domain (XII.1.10). The following second assumption is technical and we do not lose any generality with it.

Assumption II. The matrix Ao(0) has the following S-N decomposition

where Al, µ2f ... , An are eigenvalues of Ao(0) and .,V is a lower-triangular nilpotent matrix.

Note that [Ni can be made as small as we wish (cf. Lemma VII-3-3).

The following third assumption plays a key roll.

Assumption III. The matrix Ao(0) is invertible, i.e.,

In §§XII-2 and XII-3, we shall prove the following theorem.

Theorem XII-1-2. Under Assumptions I, II, and III, system (X11.1.1) has a solution

(XII.1.12) vJ = pj (x, c) (j = 1,2,... ,n)

such that

(r) pi(x,e) are holomorphie in a domain

(XII.1.13) jxl < b, 0 < lei < p, l arg el < a,

where b, p, and a are suitable positive constants such that 0 < 6 < bo, 0 <

p<po, and 0<a<ao,

(ii) p, (x, e) admit asymptotic expansions

the domain {x : lxi < 6}.

XII-2. Basic estimates

In order to prove Theorem XII-1-2, let us change system (XII-1-1) to a system of integral equations.

Observation XII-2-1. Expansion (XII.1.14) of the solution p3(x,e)

must be a formal solution of system (XII.1.1). The existence of such a formal solution (XII.2.1) of system (XII.1.1) follows immediately from Assumptions I and

III. The proof of this fact is left to the reader as an exercise.

Observation XII-2-2. For each j = 1, 2.... , n, using Theorem XI-1-14, let us construct a function g1(x,e) such that

(i) q, is holomorphic in a domain

(XI1.2.2) lxi < 6', 0 < jej < p', l arg el < a',

where 0<6'<6o,0<p'<po,and 0<a'<ao,

Denote (ql, q2, ... , qn) and (U I, U2,. .. , un) by q" and u, respectively. Then, ii satisfies the system of differential equations

Set

Then, for sufficiently small positive numbers 6, p, and -y, there exists a positive constant c, independent of j, such that for every positive integer N, the estimates

Here, BN is a positive constant depending on N and 1191.

From (XII.2.5) and (XII.2.9), it follows that

where the paths of integration must be chosen carefully so that uniformly convergent successive approximations can be defined.

Hereafter in this section, we explain how to choose paths of integration on the right-hand side of (XII.2.14).

Observation XII-2-3. Set

Without loss of generality, suppose that n real numbers w, are divided into the following two groups: