0387986995Basic TheoryC

(b)If (XIII.7.1) gives slopes of all nonvertical sides of N(L), then the operator L is factored in the following way:

(XIII.7.5)

where, for each h, N(Lh) has only one nonvertical side with slope kh.

(c)If L is factored as in (b) and each differential equation Lh[r)h] = 0 is equivalent to a system xdih = Ah(x)uh, then the differential equation (XIII.7.3) is

equivalent to x !L = diag [A, (x), A2(x), ... , Aq(x)] y'.

Statement (a) can be proved by an idea similar to the argument which is used to reduce system (XIII.6.12) to (XIII.6.14) in Case 2 of the proof of Theorem XIII-6-1.

A proof of Statement (b) is found in [Mall], [Si16] and [Si17, Appendix 1]. Look at the system

Then, Statement (c) can be verified recursively on q without any complication. The proof of Theorem XIII-7-6 in detail is left to the reader as an exercise.

Combining Observation XIII-7-1 and Theorem XIII-7-6, we obtain the following theorem.

Theorem XIII-7-7. If the distinct slopes of the nonvertical sides of N(L) are given by (XIII.7.1), then the n-th-order differential equation (XIIL7.S) has, at x =

0, n linearly independent formal solutions of the form

(XIII.7.6) rlh,v(x) = xtin.,. exp[Qh.v(x)]Oh,v(x)

where

(i) if

{(X, Yh-I + kh(X - Xh_1)) : Xh-1 C X < Xh}

is the nonvertical side of N(L) of slope kh, then

It, = Xh - Xh-1 (h = 1, 2,... , 9),

(ii) ryh,v E C, Qh,v(x) is either equal to 0 or a polynomial in x- 1/11 of the form

and the quantities Oh,v(x) are polynomials in log x with coefficients in C[[x'/']]

Here, s is a positive integer such that skh (h = 1,... , q) are integers, and PI + P2 +

+ Pq = n.

A complete proof of Theorem XIII-7-7 is found in [St].

We can construct formal solutions (XIII.7.6), using an effective method with the Newton polygon N(L). The following example illustrates such a method.

Example XIII-7-8. Consider the differential operator C = x63 - x62 - 5 - 1 or the third-order differential equation C[q] = 0. The Newton polygon N(C) is given

by Figure 6. In this case, q = 2, kl = 0, and k2 = 1.

1 3

FIGURE 6.

+00

(i) For k1 = 0, set i _ E c,,,xX+m (co 0). Then,

M=0

+00

C[q] _ { (A + m)3 - (,\ + m)2} c,,,xa+m+1

M=0

t o0

(A + m + 1)C,,,xa+m = 0.

m=o

Hence,

This implies that q = CO (x-1 - 2) is a solution of C[r7] = 0.

XIII-8. Gevrey property of asymptotic solutions at an irregular singular point

In this section we prove a result which is more precise than Theorem V-1-5 ([Mai]). In §XIII-3, we stated an existence theorem of asymptotic solutions for a given formal solution of an algebraic differential equation (cf. Theorem XIII-3-6). If differential equation (XIII.3.3) has a formal solution, we can transform (XIII.3.3) to the form

and G(x, yo, yl, ... , y.-I) is a convergent power series in (x, yo, yl ... y,,-,) (cf.

(SS31 and [Mal2j). Here, it can be assumed without any loss of generality that

(i) ah (h = 0,1, ... , n) are convergent power series in x and a # 0,

00

(ii) differential equation (XIII.8.1) has a formal solution p(x) = E a,,,x"' E

m=0

CI[x]I,

(iii)M is an integer such that for any differential operator At of order not greater than n, the two Newton polygons N(C - xMIC) and N(C) are identical (cf.

Definitions XIII-7-4 and XILI-7-5).

Using Theorem XIII-3-6, we can find

(a)a good covering {S1, $2, ... , SN } at x = 0,

(b)N solutions 01(x), 02(x), ... , ON (x) of (XIII.8.1) in S1, S2,... , SN, respec-

tively such that of are holomorphic and admit the formal solution p(x) as their asymptotic expansions as x -+ 0 in St, respectively.

Set ui = 01 - 0t+1 on Se n Sjt+1. Then, ue are flat in the sense of Poincare in sectors Se n S(+1, respectively, where Sv+1 = S1. Furthermore, if we define differential operators IQ by

If the Newton polygon N(,C) has only one nonvertical side of slope 0, then x = 0 is a regular singular point of C[iI] = 0. Therefore, in this case the formal solution p(x) is convergent (cf. Theorem V-2-7). Let us assume that./V(C) has at least one side of positive slope. In such a case, let

Theorem XIII.8.4. Let a linear differential operator L = 6 - A(x) be given, where 6 = , and A(x) is an n x n matrix whose entries are meromorphic in a

neighborhood of x = 0. Also, let (X111.8.3) be all the positive slopes of the Newton polygon N(C) of the operator C. Assume that

(1) k1 ?

51

(2) C[ f] is meromorphic at x = 0 for a f (x) E C[[xiln.

Then, there exist a finite number of directions arg x = 0 (1= 1,2,... , p) such that i f 0 34 Ot f o r I = 1 , 2, ... , p, there exist q formal power series j, (v = 1, 2, ... , q)

A complete proof of this theorem is found in [BBRS]. In this case, f is said to be {k1, k2 ... , kq }-multisummable in the direction arg x = 0. We can also prove multisummability of formal solutions of a nonlinear system. For those informations, see, for example, [Ram3], [Br), [RS21, and [Bal3j.

EXERCISES XIII

XIII-1. Using Observation XIII-6-5, diagonalize the following system:

for large )x[.

XIII-2. Show that there is no rational function f (x) in x such that

(a"f)(r) + rzf(x) = x

Hint. One method is to show that the given equation has a unique power series solution in X-1 which is divergent at x = oo. Another method is to observe that any solution of this equation has no singularity in ]x] < +oo except possibly at x = 0. Furthermore, if p(x) is a rational solution, then some inspection shows that p(x) does not have any pole at x = 0. This implies that p(x) must be a polynomial.

But, we can easily see that this equation does not have any polynomial solution

(cf. {Si20J).

XIII-3. Find a cyclic vector for the differential operator £[ 3 = x;ii + 'A, with a

XIII-6. Let A(x) be an n x n matrix whose entries are holomorphic and bounded in a domain Ao = {x : 3x! < ro} and let j (x) be a Cn-valued function whose entries are holomorphic and bounded in the domain Ao. Also, let Al, A2, ... , An be eigenvalues of A(O). Assume that det A(0) # 0. Assume further that two real numbers 01 and 82 satisfy the following conditions:

(1)01 < 02,

(2)none of the quantities Aye-'ke (j = 1, 2, ... , n) are real and positive for a

positive integer k if 01 < 8 < 02,

(3) Ape-ike' and Aqe-'k93 are real and positive for some p and q.

Show that there exist one and only one solution f = fi(x) of the system x'`+1 ds =

A(x)f+ x f (x) such that the entries of fi(x) are holomorphic and admit asymptotic

expansions in powers of x as x - 0 in the sector S = {x : 0 < !xj < r0, 81 - <

2k

argx<82+2k

Hint. If we use Corollary XIII-1-3 at x = 0, it can be shown that for each 8 in the

O(x; 8) are holomorphic and admit asymptotic expansions in powers of x as x - 0

'r

in the sectorial domain Sa = x : 0 < jxj < ro, I arg x - 81 < 2k + ea }, where Ee is

a sufficiently small positive constant depending on 8. If 10 - 8'l is sufficiently small, then ¢(x; 0) = d(x; 8).

XIII-7.

(a) Show that j (x) _ (-1)m(m!)xr+L is a formal solution of

and that f is not convergent except at x = 0.

(b)For a given direction 8, find a solution fe(x) of (E) such that fe(x) ^- j (x) as

x0 in the direction arg x = 8.

(c)Calculate fe, (x) - fe, (x) for two given directions 81 and 02.

Hint. For Part (b), use the following steps:

Step 1. Apply Theorem XIII-1-2 to the given differential equation. To do this, we

must change x = 0 to t = oo by x = 1. Then, the differential equation becomes

du = y - t 1. In this case, n = 1, r = 0, and the eigenvalue is ,u = 1. Set arg u = 0.

Then, the domain D(N, ry) (cf. (XI 11. 1. 15)) is

D(N,7) = {t:iti > N, I argt - 2gir! < 32 -

f}