0387986995Basic TheoryC

398 XII. ASYMPTOTIC SOLUTIONS IN A PARAMETER

Ao. Assume also that f(t, y, e) and the coefficients fm(t, y) are periodic in t of

dy

period 1. Show that the differential equation dt = 27riy + e[y + ef (t, y, e)] has a

periodic solution U = e) of period 1 such that

(a) the entries of (t, e) are holomorphic with respect to (t, e) in a domain D =

{ (t, e) : I `3'tJ < d, -oo < Rt < +oo, 0 < JeJ < r, I arg el < a} for some (d, r, a)

such that0<d<do,0<r<r0,and0<a<ao,

00

(b) j(t, e) ^ - - E em&m(t) as a -+ 0 in the sector S = If : 0 < Jej < r, J arg el <

M=0

a}, where coefficients pm(t) are holomorphic and periodic of period 1 in the domain D' = It : J:33tI < d, -oo < Rt < +oo}.

Hint.

Step 1. Construct the solution +li(t, c", e) to the initial-value problem

Substep 1. First, construct a formal solution

+00

+G(t, , e) _ E em'+Gm(t, l = e2x1cd + ete2attC + O(e2).

M=0

The coefficients 1& (t, a) are holomorphic in a domain

S2o = {(t, cl : 13'tI < do, -oo < Rt < +oo, Icl < ryo),

where -yo is a positive number. To prove this, set y = e2i't(c + CZ-) to change (IP)

to

The function f (t, e2x't (c" + ez-),e) admits an asymptotic expansion in powers of e (cf. Proof of Theorem XI-1-8) whose coefficients are polynomials in the entries of the vector z. Coefficients ci can be calculated successively.

Substep 2. Show that iP(t, c", e) V;(t, c, e) as e -. 0 in a sector S = {e : 0 < je1 < r, I argel < a} uniformly in a domain 1Y = {(t,c) : J)tj < d, IRtI < T, Icl < ry}, where T is an arbitrary positive number and -y > 0 depends on T. In this argument, the Gronwall's inequality (Lemma 1-1-5) is useful.

Step 2. Solve the equation 1 (1, cE, e) = E. This equation has the form

where h(c", e) admits an asymptotic expansion as a -' 0 in the sector S uniformly for Icd < y. To solve this, we can use successive approximations as follows:

4W = 0, Ch(e) = e46y_1(e),e).

Then, the approximations 64(e) converge to a limit c(e). Using Theorem XI-1-12, we can conclude that ee) admits an asymptotic expansion in powers of e.

Step 5. Using again Steps 1 and 2, find a function r(x, e) such that

(a)r(x, e) is holomorphic and bounded for jxj > R' > 0 and e E S,

(b)r(x, e) = 0 as a - 0 in S uniformly for jx[ > R',

(c)for a sufficiently large M' > 0, the transformation v = [I +x-(h1'+1)r(x,e)jw changes (s") to

N M

dw = xk ` x-mam(f) + x-(N+1) E ? bm(e) w,

Lm=o

M-

whereb,,,(e)-0ase-0 inSform>k+1.

XII-13. Let A(t) be a 2 x 2 matrix whose entries are holomorphic in a disk It : it[ < po} such that the matrix

(M) P ()' {kA(!)P(1) -P (-X1)1

is not triangular for any 2 x 2 matrix P(t) such that the entries of P(t) and P(t)-1 jtj < po}. Show that there exists such a 2 x 2

matrix P(x) for which matrix (M) has the form xk B Gl ) with a 2 x 2 matrix B(t)

wh ose entries are polynomials in t and whose degree in t is at most k + 1.

Hint. See [JLP]. This result can be generalized to the case when A(t) is a 3 x 3 matrix (cf. [Ball) and [Bal2J).

XII-14. Let P(t) be a 2 x 2 matrix such that the entries of P(t) and P(t)-1 are

holomorphic in a disk V = it : jtj < po} and that the transformation y" = P (!) u"

X

changes the system

with a 2x 2 matrix B(t) whose entries are polynomials in t. Show that the degree of the polynomial B(t) is not less than 2. Also, show that there exists P(t) such that the degree of the polynomial B(t) is equal to 2.

Hint. To prove that there exists P(t) such that the degree of the polynomial B(t)

suitable constants a, Q, y, and b. To show that the degree of the polynomial B(t) is not less than 2, assuming that the degree of the polynomial B(t) is less than 2, derive a contradiction from the following fact:

XII-15. Let A(t) be a 2 x 2 triangular matrix whose entries are holomorphic in a disk it : Its < po}. Show that there exists a 2 x 2 matrix Q(x) such that

(i)the entries of Q(x) and Q(x)-I are holomorphic for jxi > o and meromorphic at x = oo,

xkB 1 f u" with a 2 x 2 matrix B(t) whose entries are polynomials in t and

whose degree in t is at most k + 1.

Hint. See IJLP]. This result can be generalized to the case when A(t) is a 3 x 3 matrix (cf. [Ball] and [Bal2j).

Assumption I. Each function f j (x, V-) (j = 1, 2,... , n) admits a uniform asymp- totic expansion

where coefficients fjk(v-) are holomorphic in the domain

Observation XIII-1-1. Under Assumption I, fo(x), ajh(x), and fl,(x) admit asymptotic expansions

as x oo in sector (XIII.1.5) with coefficients in C. Let A(x) be then x it matrix whose (j, k)-entry is ajk(x), respectively (i.e., A(x) = (a,h(x)). Then, A(x) admits an asymptotic expansion

as x - oo in sector (XIII.1.5), where Ak = (ajhk). The following assumption is technical and we do not lose any generality with it.

Assumption II. The matrix A0 has the following S-N decomposition:

where µ1,µ,2, ... , µ are eigenvalues of Aa and N is a lower-triangular nilpotent matrix.

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

The following assumption plays a key roll.

O(x) are holomorphic and admit asymptotic expansions in powers of x-1 as x -+ 00

in the sector S = {x : IxI > Ro, I argx - 01 < Zk + e}.

Proof.

The main claim of this corollary is that the asymptotic expansion of the solution is valid in a sector I argx-8 < 2k +e whose opening is greater than k So, we look

at the sector D(N, y) of Theorem XIII-1-2. In the given case, ao = +oo, r = k -1, and µ, = Aj (j = 1,... , n). The assumptions given in the corollary imply that

for a sufficiently large Ro > 0 and a sufficiently small e > 0 if we use xe-i8 as the independent variable instead of x.

XIII-2. Basic estimates

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

Observation XIII-2-1. Expansion (XIII.I.17) of the solution pj (x)

must be a formal solution of system (XIII.1.1). The existence of such a formal solution (XIII.2.1) of system (XIII.1.1) follows immediately from Assumptions I and III. The proof of this fact is left to the reader as an exercise.

Observation XIII-2-2. For each j = 1,2,... , n, using Theorem XI-1-14, let us construct a function z3(x) such that

(i) z j (x) is holomorphic in a sector

as x - oo in sector (XIII.2.2), respectively.

Consider the change of variables

as x -+ oo in sector (XIII.2.2). Thus,

as x -' oo in sector (XIII.2.2).