0387986995Basic TheoryC

380 XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

To show (iii), from (XII.2.11), (XII.3.3), and Lemma XII-2-4, we obtain

Iu-(h+l)(x,E)-u-(h)(x,E)I :5c8 sup Iiz(h)(t,E)-tL(h-1)(t,E)I tED(6)

in (XII.3.5) for h = 1 , 2, 3, .... Hence, the sequence (u -1 h ) (x, E) : h = 0,1, 2, ... } is convergent to (x, f) uniformly in (XII.3.5). Furthermore, uj = Oj (x, e) (j = 1, 2,... , n) satisfy system (XII.2.14). Thus, the proof of Theorem XII-1-2 is completed. 0

XII-4. A block-diagonalization theorem

Consider a system of linear differential equations

where a is a positive integer, y E C", and A(x, c) is an n x n matrix. The entries of A(x, E) are holomorphic with respect to a complex variable x and a complex parameter c in a domain

(XII.4.2) lxi < ao, 0 < IEI < po, I argEl < no,

where bo, po, and ao are positive numbers. Assume that the matrix A(x, e) admits a uniform asymptotic expansion in the sense of Poincar6,

Suppose that Ao(O) has a distinct eigenvalues A1, A2i ... , At with multiplicities

Here, I,, is the nj x n, identity matrix and Yj is an n, x nj lower-triangular nilpotent matrix. The main result of this section is the following theorem.

Theorem XII-4-1 ([Si7]). Under assumptions (X11.4.8) and (X11.4.6), then exists an n x n matrix P(x, e) such that

(i) the entries of P(x, e) are holomorphic with respect to (x, e) in a domain

where 6, p, and a are positive numbers such that 0 < 5 < 60, 0 < p < po and

0<or <ao,

(ii) P(x, e) admits a uniform asymptotic expansion

where the entries of coefficients P,(x) are holomorphic with respect to x in the domain

where B , (x, e) is an n, x n, matrix (j = 1, 2, ... , e),

(iv) the matrix B, (x, e) admits a uniform asymptotic expansion

382 XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

Remark XII-4-2.

(a) Set

(XII.4.16) diag [B1v(x), B2 ,(x), ... , Bl,(x)] -

Then, the coefficient matrix B(x, e) of system (XII.4.13) admits a uniform

0

asymptotic expansion B(x, e)

'=a

(b)When a fundamental matrix solution Z(x, e) of (XII.4.13) is known, a fundamental matrix solution Y(x, e) of (XII.4.1) is given by Y(x, e) = P(x, e)Z(x, e).

(c)In the case when the matrix Ao(O) has n distinct eigenvalues, by Theorem

XII-4-1, we can diagonalize system (XII.4.1).

(d)In the case when eigenvalues of the matrix Ao(O) are not completely distinct, the point x = 0 is, in general, a so-called transition point. In order to study

behavior of solutions in the neighborhood of a transition point, we need a much deeper analysis of solutions of system (XII.4.1). For these informations, see, for example, [Wash], [Was2], and [Si12].

Proof of Theorem XII-4-1.

We prove this theorem in two steps. The proof is similar to that of Theorem

VII-3-1.

Step 1. We show that there exist a positive number 6 (< oo) and an n x n matrix Po(x) such that

(i)the entries of Po(x) and Po(x)-1 are holomorphic in the domain {x : JxI < b} and Po(0) = In,

(ii)the matrix Co(x) = Po(x)-1Ao(x)Po(x) is in a block-diagonal form

(XIL4.17) Co(x) = diag [Col)(x),Co2)(x),... ,C(c)(x)]

where Ca) (x) is an n, x nj matrix such that

In fact, two matrices Po(x) and Co(x) must be determined by the equation

and

B(x, e) = diag [b(') (X, E), b(2) (X, E), . . . , B(') (x, E)] ,

where A(jk) (x) and P(Jk) (x) are nj x nk matrices and BJ (x, c) is an nj x nJ matrix

(j, k = 1,... , e). Furthermore, set

Then, equation (XII.4.23) becomes

l

B(.)(x, f) = A(Jj)(x,E) + Ey: A(Jh)(x,E)P(hJ)(x,e)

(XII.4.26)

h=1

(j = 1,2,...,t)

and

(XII.4.27)

h=1

Replacing P(jk)(x,E) by an nJ x nk matrix X(Jk), we derive a system of nonlinear differential equations

a system of nonlinear differential equations. Upon applying Theorem XII-1-2, we

construct an analytic solution of this nonlinear system in a domain (XII.4.8) admitting an asymptotic expansion in e as described in Theorem XII-4-1. This completes the proof of Theorem Xll-4-1. 0

XII-5. Gevrey asymptotic solutions in a parameter

In this section, using Theorems XI-2-3 and the proof of Theorem XII-1-2, we construct Gevrey asymptotic solutions of a system of differential equations

where x is a complex variable, y E Cn, E is a complex parameter, a is a positive integer, and Ax, if, e) is a Ca-valued function of (x, y', f). Define three domains by

A(So) = {x E C : IxI < bo}.

(XII.5.2)

S(ro, no) = {E E , arg E < ao, 0 < H < ro}.

Also, define two matrices A(x, E) and Ao(x) by

89

Oyn (x, 0, f)

8y1(x,E)

and

respectively.

We first prove the following lemma.

Lemma XII-5-1. Assume that

(i)Ax, y", e) is holomorphic with respect to (x, f, c) in a domain A(bo) x Q(po) x S(ro, no), where 6o, po, ro, and no are positive numbers,

(ii)f (x, y", c) is bounded on A(60) x i2(po) x S(ro, no),

(iii)the matrix Ao(x) defined by (XII. 5.4) exists as e - 0 in S(ro, no) uniformly in x E A(6o) and Ao(0) is invertible,

(iv)f (x, 0, e) is flat of Gevrey order -r as e -+ 0 in S(ro, no) uniformly in x E

A(60), where r is a non-negative number.

Then, there exist three positive numbers b, r, and or such that system (XIL5.1) has a solution O(x, e) which is holomorphic in (x, t) E A(6) x S(r, a) and that (x, c) is flat of Gevrey order -r as e -' 0 in S(r,a) uniformly in x E L1(6).

386 XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

Proof.

Note first that

(XII.5.5) I exp1- k1I = expJ-cJfJ-'cos(k(arge))]

for any positive numbers c and k. Note next that assumptions (i) and (iv) imply th in the case when r = 0, Ax, 0, e) is identically equal to 0 for (t, f) E A(b) x S(r, a).

Hence, in this case, 0 is a solution of (XII.5.1). In the case when r > 0, it holds that

for some positive numbers K and c if (x, e) E A(b) x S(r, a) for sufficiently small positive numbers 6, r, and a and if

k

(XII.5.7)

(cf. Theorem X1-3-2). Therefore, (XII.5.5) and (XII.5.6) imply that

Then,

exp[cE-k1f(x, eXp[-cE-k]u, e)

Using a method similar to the proof of Theorem XII-1-2, we construct a bounded solution u = tlr(x, e) of (XII.5.11). Therefore, system (XII.5.1) has a solution of the form y" _ ¢(x,e) = expl-ce-k]y,(x,e). This completes the proof of Lemma

XII.5.1.

The main result of this section is the following theorem, which was originally conjectured by J: P. Ramis.

Theorem XII-5-2. Assume that

(i)f (x, y', e) is holomorphic in (x, y, e) on a domain 0(6o) x f2(po) x S(ro, ao), where bo, po, ro, and ao are positive numbers,

(ii)f (x, y, e) admits an asymptotic expansion F(x, y, e) of Cevmy orders as a --+ 0 in S(ro, ao) uniformly in (x, y-) E A(bo) x f2(po), where s is a non-negative

number,

(iii)the matrix Ao(x) given by (XIL5.4) is invertible on 0(bo),

(iv)it holds that

c-

with coefficients p"t(x) which are holomorphic on A(bo),

(2)there exist three positive numbers 6, r, and a such that (XII. 5.1) has an actual solution ¢(x, e) which is holomorphic in (x, e) E A(5) x S(r, a) and that ¢(x, e) admits the formal solution P(x, e) as its asymptotic expansion of Gevrey order

max { 1, s } as e - 0 in S(r, a) uniformly in x E L1(b).

a

J11

Proof.

If a positive number & is sufficiently small, for every real number 6, there exists a C"-valued function fe(x,y,e) such that

(a)fe(x, y', e) is holomorphic in (x, y", e) on a domain A(60) x f2(po) x Se(ro, &), where

(XIL5.14) Se(ro,&) = {e: jarge - 61 < &, 0 < fej < ro},

(b)fe(x, y, e) admits an asymptotic expansion F(x, y e) of Gevrey order s as e - 0 in Se(ro, &) uniformly in (x, y) E 0(bo) x f2(po), where s is the non- negative number given in Theorem XII.5.2.

Such a function fe(x, y, e) exists if & is sufficiently small (cf. Corollary XI-2-5). In particular, set

such that &(x, e) is holomorphic and bounded in (x, e) E i (b1) x Seri, al) for suitable positive numbers 61, r1, and al. Using Theorem XII-1-2, it can be shown that such a solution of (XII.5.16) exists.