Differential Equations & Asymptotic Theory In Mathematical Physics: Wuhan University, Hubei, China 20 - 29 October 2003 (Series in Analysis)

Differential Equations 81 Asymptotic Theory i n Mathematical Physics

SERIES IN ANALYSIS Series Editor: Professor Roderick Wong City University of Hong Kong, Hong Kong, China

Published Vol. 1

Wavelet Analysis edited by Ding-Xuan Zhou

Vol. 2

Differential Equations and Asymptotic Theory in Mathematical Physics edited by Hua Chen and Roderick S.C. Wong,

Vol. 2

Differential Equations & Asymptotic Theory i n Mathematical Physics 20 - 29 October 2003

Wuhan University, Hubei, China

Editors

Chen Hua Wuhan University, China

Roderick Wong City University of Hong Kong, Hong Kong

K@World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG K O h

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PREFACE

This book consists of lecture notes for four mini courses delivered at the Conference on Differential Equations and Asymptotic Theory in Mathematical Physics held in Wuhan University, China in October, 2003. Each course contains five one-hour lectures. These notes give the readers a very good idea about recent developments in the area of asymptotic analysis, singular perturbations, orthogonal polynomials, and application of Gevrey asymptotic expansion to holomorphic dynamical systems. It also includes invited papers presented at the conference. The topics include the asymptotic behaviour for the formal solutions of the singular partial differential equations in complex domains and the applications of the partial differential equations in compound crystals, pricing of real options, hydraulic engineering, diblock copolymer, etc. The conference has more than one hundred participants. We take this opportunity to thank all of them, and in particular, the authors of the articles included in this book.

The Editors Hua CHEN Roderick S.C. WONG

V

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CONTENTS

Preface

..................................

v

PARTI MINI-COURSES Ismail, Mourad E. H. Lectures on Orthogonal Polynomials . . . . . . . . . . . . . . . . . 1 Ramis, Jean-Pierre Gevrey Asymptotics and Applications to Holomorphic Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . .

44

Ward, Michael J. Spikes for Singularly Perturbed Reaction-Diffusion Systems and Carrier's Problem . . . . . . . . . . . . . . . . . . . 100 Wong, Roderick S. C. Five Lectures on Asymptotic Theory . . . . . . . . . . . . . . . . 189

PARTI1 INVITED PAPERS Bohun, C. S., F'rigaard, I., Huang, H.X. and Liang S. Q. A Perturbation Model for the Growth of Type 111-V Compound Crystals . . . . . . . . . . . . . . . . . . . . . . . . .

263

Chen, H. and Yu, C. Asymptotic Behaviour of the Trace for Schrodinger Operator on Irregular Domains . . . . . . . . . . . . . . . . . . . 280 Jiang, L. S. and Ren, X. M. Limitations and Modifications of Black-Scholes Model

. . . . . . 295

Li, T.T. (Li, Daqian) Exact Boundary Controllability of Unsteady Flows in a Network of Open Canals. . . . . . . . . . . . . . . . . . . . . . . vii

310

...

Vlll

Miyake, M. and Ichinobe, K. Hierarchy of Partial Differential Equations and Fundamental Solutions Associated with Summable Formal Solutions of a Partial Differential Equation of non Kowalevski Type . . . . . . . 330 Tahara, H. On the Singularities of Solutions of Nonlinear Partial Differential Equations in the Complex Domain, I1 . . . . . . . . . 343 Tan, Y . J. and Chen, X. X. Identifying Corrosion Boundary by Perturbation Method . . . . . 355 Wei, J. C. Existence and Stability of Lamellar and Wriggled Lamellar Solutions in the Diblock Copolymer Problem . . . . . . . . . . . 365

LECTURES ON ORTHOGONAL POLYNOMIALS

MOURAD E. H. ISMAIL* Department of Mathematics University of Central Florida Orlando, FL USA 32816 E-mail: [email protected]

These lecture notes contain a brief introduction to orthogonal polynomials and mention some applications which are not readily available in books on special functions and orthogonal polynomials. Since these lectures were prepared for a conference on differential equations and asymptotical theory in mathematical physics, we naturally emphasized differential equations satisfied by orthogonal polynomials and attempted to explain the role asymptotics play in the theory of orthogonal polynomials. Due to space and time limitations we have not treated orthogonal polynomials on the unit circle, nor have we touched the techniques of obtaining asymptotics of large zeros of special functions or largest and smallest zeros of orthogonal polynomials. The article by Rod Wong 65 in these proceedings complements this article and provides details of asymptotic techniques.

1. Construction of Orthogonal Polynomials Given a probability measure p supported on a subset of R and having finite moments, Jwz n d p ( z ) , we wish t o construct a sequence of polynomials {p,(z)} such that (i) p n ( z ) has precise degree n for all n, p,(z) = y,zn+ terms, 7, > 0, (ii) J,P?n(z)Pn(z)dP(z) = &Tl,n.

lower order

The Gram-Schmidt procedure shows that { p , ( z ) } exists and is unique. Let

*Research supported by NSF grant DMS 99-70865. 1

2

and define the Hankel determinants

It is clear from (1.1) that the quadratic form xIk=O&+kXj?i?k is IC:zjzj12dp(z), hence is positive definite. Thus D, > 0 for all

sw

n , n = O , l , ....

Theorem 1.1. The polynomials p n ( x ) have the form Po

PI

.. .

Pn-1

1

P1 ... Pn PZ ... Pn+1

.. .

Pn 2

...

...

PZn-1 Zn

Proof. Denote the right-hand side of (1.3) by q n ( x ) . Thus qn(z)=

/%

zn

+ lower order terms,

whence the leading coefficients in q n ( z ) is positive, For k

...

< n,

. ..

which is zero. If k = n then the right-hand side of the above equation is = Therefore

D n / d G d=.

) w(z) and w is called When p is absolutely continuous, we write ~ ' ( z= a weight function.

3

Exercise. Recall the gamma integral

Let p be absolutely continuous with p ' ( x ) = x " e - " / r ( a

+ 1). Prove that

where

Moreover

Note t h a t we proved that

Theorem 1.2. [Heine]. The orthonormal polynomials have the integral representation

Proof. In the representation (1.3) write the entries p k , p k + l , . . . , pk+n in 1 as J, Xk+,dp(Xk+l), . . . J, X ~ ~ ~ d p ( X k + l for ), 0 I k < n. Thus

row k

+

4

dGp,(3:) has the representation

1

1

3:

...

2"

Recall the evaluation of the Vandermonde determinant

Xi).

Let a be a permutation on 1,. . . ,n and S, denote the symmetric group. In the above integral we can replace X I , . . . ,A, by Xo(l), . . . , Xo(,) then rearrange the rows in the determinant to have XI,. . . ,A, in rows 1 , 2 , . . . , n. This produces a multiplicative factor sign (a), sign (a) being the sign of the permutation a. By summing over a E S, and dividing by the order of the symmetric group (= n!)we see that d z p , ( 3 : )is

Corollary 1.3. The Hankel determinants have the representation

5

Proof. Equate coefficients of xn in (1.6).

0

Remark 1.4. Let {$,(x)} be a sequence of polynomials such that 4, has precise degree n. One can express p n ( z ) as a linear combination of 40(x), 41 (X), . . . ,4n(x)in the following way

with

and

-

D, = det(ai,j), 1 5 i , j 5 n.

(1.10)

The determinant representations for orthogonal polynomials seem to be underused by experts in the area. For a clever use of these determinants see Wilson 64. We next show how Heine’s formula appears naturally in random matrices, see Mehta 42. All matrices considered here are n x n matrices. Let M be a Hermitian matrix and let M = UAU*, where A is a diagonal matrix (formed by the eigenvalues of M ) and U a unitary matrix. Here U = ( u j k ) , U* = (u;~), where 2 ~ = ; ~ujk. The set of all Hermitian matrices is isomorphic to RnZbecause a Hermitian matrix has n real numbers on the diagonal and n(n - 1)/2 complex numbers on the super diagonal. Thus a Hermitian matrix depends on n + 2[n(n- 1)/2] = n2 real entries. Let V be a polynomial, and let tr(M) denote the trace of M (= the sum of its diagonal elements). The trace is invariant under similarity, so tr(M) = tr(UAU*) = tr(A). Because M j = (UAU*)j = U A j U * , then tr(V(M)) = tr(V(h)) = Cj”=, V(Aj). We consider the integrals

I ( F )=

J

F (M)e-t‘(V(M)) dM,

(1.11)

MZHermitian

where dM is the Euclidean measure on the space of Hermitian matrices (= I t n z ) . If we change the variables in the upper triangular part of M to the eigenvalues XI,. . . ,A, of M and the elements of U , the Jacobian will be

6

nlli<jln(Ai - Aj)2, see Deift l7 for this calculation. We now assume that F ( M ) depends only on the eigenvalues of A4 and is a symmetric function of the eigenvalues. Perform integration on the U parameters to obtain

I ( F ) = Constant

Ln

n n

JJ,

F ( x ~.,. . ,A,>

- Xk)'

e-v(x3)d~s.

s=l

l<j 0. I f j < n - l t h e n

7

because p j ( x ) x has degree j

+ 1 which is less than n, It is also clear that

= d,-l,n = 3;~-1/7,. Moreover d,,, Thus d,,,-i dn,, E R. Thus we proved the following result.

= J w x p i ( x ) d p ( x ) ,SO

Theorem 2.1. Let {p,(x)} be orthonormal with respect to p . {p,(z)} satisfies the three t e r m recurrence relation

+

+

xPn(x) = a n + l ~ n + l ( x ) b n ~ n ( x ) a n p n - l ( x ) , with b, E

Then (2.2)

R and a, > 0. Moreover an

= yn-l/yn =

@ZZIDn-1.

(2.3)

Another important normalization is the monic form

Pn(X) = xn + lower order terms.

(2.4)

Thus

In this normalization the recurrence relation (2.2) becomes

.P,(.)

+ b,P,(z) + .2n(z).

= P,+I(Z)

(2.6) A consequence of (2.2) is the following theorem, see Chapter 2 in Szeg6 58

Theorem 2.2. The zeros of p,(x) are real and simple and belong to the smallest interval containing the support of p . Moreover the zeros of p,(x) and p,+l(x) interlace. One source of information on classical orthogonal polynomials has been that they satisfy a Sturm-Liouville problem. Consider the eigenvalue problem

for x E ( a ,b) and y E L 2 ( a ,b; w). It is assumed that w(z) > O , p ( x ) > 0 on ( a ,b ) , w and p are continuous on ( a , b ) with w(a+) = w ( b - ) = 0 . Further we assume.that J , b IQ,(x)lw(x) d x < 00. The eigenvalues of this symmetric eigenvalue problem are all real. If A1 # A2 then (2.8)

8

The Jacobi polynomials { P p ’ P ’ ( z )satisfy }

= n(n

+ a + p + l)PpP’(z),

(2.9)

where

we,p(z) = (1 - z y ( l

+ zy,

z E [-1,1].

(2.10)

The ultraspherical polynomials correspond t o the case a = p. The Legendre polynomials occur when a = ,B = 0, while the Chebyshev polynomials of the first and second kinds correspond t o a = p = respectively. The Laguerre polynomials {L,(0) (z)} arise when 3: is scaled as -1+2z/a then we

~ i ,

let a + 00. The Hermite polynomials arise when a = /3 and z is replaced by z / a and we let a -+ co. As an example of a classical polynomial we derive some of the properties of Laguerre polynomials. We require the use of the Chu-Vandermonde sum, Andrews, Askey, Roy

(2.11)

The weight function is

w,(z) = zae-z/I’(a

+ l),

z

> 0.

(2.12)

Let the polynomials be L p )(x), (2.13)

where { c , , k } are coefficients to be determined. The factor sum terminate at k = n, since

(-n), = 0,

m > n.

(-n)k

makes the

(2.14)

9

We first evaluate m < n. Clearly

Jr

xmL?)(x) e-"x"dx and demand that it vanishes for

To evaluate the above sum we take C,,k = C , / ( A ) k in order t o apply (2.11). Thus we want, for m < n,

=.c n

0

k=O

+ + 1)k -- ( A - m - a - 1 ) n

m k!(A)k

(-n)k(a

From (2.14), it follows that A = a

+ 1 will work. Thus (2.15)

where c, was chosen a.s ( a f l ) n / n ! .This normalization has been adopted for over a hundred years and { L ? ) ( x ) } are neither orthonormal nor monic. Observe that the above calculations establish

=

(-iyr(a + n + 1).

To find J r ( L ? ) ( x ) ) 2 x " e - " d x , we proceed as follows, where use the above integral evaluation.

lw

x"e-"(L?)(x))2 d x

10

Therefore

(2.16)

It is important to note that in the above derivation the only integral evaluation used is the gamma function integral. When we expressed L?)(x) as a linear combination of xk we used the same integral that evaluated the total mass of the measure, namely xLYecxdx, with (Y replaced by a+a+k. In other words xnfk attaches nicely to xae-x. This attachment procedure has been very useful in constructing closed form expressions for orthogonal polynomials, see Andrews and Askey and Berg and Ismail lo. To find the three term recurrence relation satisfied by Lp)(x) we first observe that the coefficient of xn in Lp)(x) is (-l)n/n!. Hence xLp)(x) (n is a polynomial of degree at most n. Let

'

+

+ l)L?:)(x)

-rcLp)(x) = ( n + l)L?Jl

+ BnL?)(x) + C,L?2'(2).

By equating coefficients of xn and xn-' on both sides of the above equation and applying (2.15)we establish the recursion relation

-xLp) (x)= ( n+ l)L?il (x)- ( n+ a+ 1)Lp)(x)+ ( n+ a ) L f l (x). (2.17) In

3 3 we shall establish the differential equation d -.--pyx) dx

d + ( a+ 1 - x)-Lp)(x) + nLp)(x) = 0. dx

(2.18)

The Christoffel-Darboux formula

3. Differential Equations In this section we derive differential equations and raising and lowering operators for general orthogonal polynomials when

dp = w(x)dx, and

W(X) = e-"("),

x

E

Theorem 3.1. Assume w' is continuous on (a,b) and

( a ,b).

(3.1)

11

exists for nonnegative integers n. Define A n ( x ) , B n ( x ) b y

(3.3)

and assume that the boundary terms in (3.2)-(3.3) exist. Then d -dx~ n ( x ) = An(x)pn-l(x)- Bn(x)pn(x).

(3.4)

Theorem 3.1 follows from the Christoffel-Darboux identity (2.18) and integration by parts. The details are in 15. Theorem 3.1 is due to Bauldry ’, Bonan and Clark 1 1 , Mhaskar 43, and Chen and Ismail 15. Define operators L1,n and L s , by ~

d L1,n = - B n ( x ) , L2,n = dx Thus we write (3.4) as

+

d -z + B n ( x )+ ~‘(x).

L1,npn( x )= An (x)pn-1 ( x ) ,

(3.5)

(3.6)

hence L1,n is a lowering or annihilation operator for {pn(x)}.By eliminating pn-l(x) between (3.4) and (2.2) we establish

Thus L2,, is a raising or creation operator. In fact L I , and ~ Ls,~ are adjoints with respect to the inner product

(f,9) = see Chen and Ismail

15.

/

b

-

f(x)g(x)w(Wx,

Combining (3.6) and (3.7) we arrive at

which is a second order differential equation satisfied by the polynomials. One important application of the differential equation (3.8) is that one can apply the Liouville-Green (or WKB) approximation and derive the large

12

n asymptotic behavior of pn(x). One can also derive uniform asymptotic expansions. For details see R. Wong 's article in these proceedings 65.

Example 1. From (2.14)-(2.16) we see that the orthonormal Laguerre polynomials are

=

qa+ iyX.

Similarly, integration by parts and the orthogonality relation yield

- -n

'Yn -

'Yn- 1

Therefore

Bn(z) = -n/z.

A,(x) = d = / x , In this case (3.4) leads to d dx

-X-LP)(X)

= (n

+a)Lpl(x)

-

nL?)(z).

The differential equation (3.8) becomes d2 dx2

z-Lp(x)

+ (1+ a - X ) -dxdL P ) ( X ) + nL?)(x)

which is of Sturm-Liouville type.

= 0,

(3.9)

13

Example 2. Consider the weight function CeFX4on W,so u = x4-In C. The boundary terms in (3.2)-(3.3) vanish and we have

Since w is an even function, p,(-x) = (-l),p,(x) so pi(x) is an even function. Moreover (2.2) implies b, = J,xpi(z)w(x)dx = 0. Thus the right-hand side of the above equation is

and we find

&(x) = 4a,(x2

+ a:

(3.10)

+a;+,).

Similarly

B,(x)= 4a;x.

(3.11)

Therefore

pL(x) = 4a,(x2

+ + ai+l)p,-l(x)

-

4aixp,(~),

(3.12)

and

= 16ai(x2

+ + ~ i - ~ ) p , ( ~ ) .(3.13)

The differential equation (3.13)was first derived by Shohat 54 through a complicated procedure. For applications see Nevai 47. One can also derive nonlinear relations among the recursion coefficients {a,} from (3.12)by using the recurrence relation

x p n ( x ) = an+lpn+l(x)

+ anPn-l(X).

(3.14)

+ lower order terms.

(3.15)

Clearly (3.14) yields

xn pQ(x) = a1ag.. .a,

+ C,Z,-~

Equating the coefficients of x*-l in (3.12)proves

+

n - 4 4 a i d+l) 4a,(c,-1 a l . . .a, a1 . . . a,-1

--

+

- a,c,).

(3.16)

14

On the other hand, substituting from (3.15) into (3.14) and equating the coefficients of xn-' we find

Therefore (3.16) becomes

+ + a2+,).

n = 4 ~ 2 ( a i - ~a:

(3.17)

Many additional nonlinear relations follow from equating coefficients of other powers of x. Freud seems to have been the first to discover such nonlinear equations for recursion coefficients of polynomials orthogonal with respect to exponential weights, W(Z) = exp(-z2"). He only studied the case m = 2. Formulas like (3.17) are instances of the string equation in Physics. It is clear that one can derive nonlinear relations similar to (3.17) when v = x6+ constant using the same technique. References t o the literature on this are in 47. The treatment presented here is from 15. Chen and Ismail l 5 analyzed the case when v is a polynomial and described the corresponding A,(x) and Bn(x) functions. Qiu and Wong 49 used the differential equation and the Chen-Ismail analysis to derive large n uniform isymptotics for p,(x). The operators L1,n and Lz,, generate a Lie algebra where the product of two operators A and B is the Lie bracket [A,B]= A B - BA. Finite dimensional Lie algebras are of interest. When u = x 2 + constant, L1,n and Lz,, generate a three-dimensional Lie algebra called the harmonic oscillator algebra, Miller 45. Miller 44 characterized all finite dimensional Lie algebras that are generated by first order differential operators. Chen and Ismail l 5 proved that when u ( x ) is a polynomial, ~ ( x=)e-"("), then L I , and ~ Lz,~ generate a Lie algebra of dimension 2 m 1, 2 m being the degree of u(x). The converse is not known so we state it as a conjecture.

+

Conjecture. If the Lie algebra generated b y L I , and ~ Lz,, is finite dimensional and the support of v is I%, then the Lie algebra has dimension 2 m + 1 and u must be a polynomial of even degree. The differential equation (3.8) when expanded out becomes P;(X)

+ AL(x)/An(x)l~L(x) + sn(x)Pn(x)= 0,

- [~'(x)

(3.18)

where

(3.19)

15

4. Electrostatic Equilibrium Problems We will study Coulomb interactions in the plane where the force is 2elea/r, el, and e2 being the charges of two particles and r the distance between them. This arises in R3 if we have infinite uniformly charged wires perpendicular to the plane under consideration and el, and e2 are the charges per unit length. The potential energy is -2ele2 In r . Now consider n movable charged particles each carrying a unit charge. The particles are restricted to a subset E of R in the presence of an external field with potential energy V(x). Let 21, . . . ,x, denote the positions of the particles and assume x1 > x2 > . . . > 2,. In what follows x will denote the vector (XI,22,. . . ,xn). The total energy E ( x ) of this system is n k=l

l 0. The Legendre polynomials correspond to v = 1/2. Let

c

M _.

G(z, t ) =

C;(x)tn

(5.5)

n=O

Multiply (5.4) by tn and add for n = 1,2,. . . . In view of (5.3) and after replacing atn by tatt", we establish the differential equation

2vxG(x, t )

+ 2xt&G(x, t ) = &G(x,t ) + 2 v t G ( ~t,) + t2&G(x,t ) .

Therefore

dtG(x,t) - ~ V (-Zt ) 1 - 2xt t 2' G(x, t )

+

hence G(x,t) = f ( x ) ( l - 2xt +t2)+'. Now f(x) = 1 because 1 = Cg(x) = G(x, 0) = f(x). This proves

c 00

n=O

c,(x)tn = (1 - 2xt + t y .

(5.6)

21

To justify the above formal steps we start with the answer and reverse the steps until we reach (5.3)-(5.4).

xrfnt",

Theorem 5.1. (Darboux's Method). Let f ( t ) = g(z) = gntn be analytic in It1 < r and f - g be continuous on It1 5 r . Then

xr

f" - gn = o(?--").

(5.7)

This theorem follows from the Riemann-Lebesgue lemma, see Olver 48. Given f , the function g is called a comparison function and it normally consists of the singular part o f f .

Example. Consider the Legendre Polynomials. Their generating function is

c 00

P,(z)t" = (1 - 2zt + t

y 2 ,

(5.8)

n=O

since the Legendre polynomials correspond t o v = 1/2 in {C,"(z)}. The notation P,(z) is standard in the literature but Pn is not monic. The t-singularities of the generating function (5.8) are at t = a , t = p, where

a,p=zfdz2-1, Here z E C. Note that a and only if z E [-1,1].

= ,B if

and

IPI5IaI.

and only if z = fl.Moreover la1 =

IpI if

Case I: z = f l . In this case the right-hand side of (5.8) is (1~ t ) - 'hence , Pn(fl)= (fl)"and no asymptotic analysis is needed. Case 11: z E C \ [-I, 11. In this case

IPI < la[.Let

where we used the binomial theorem in the last equation. It is easy to see that C," Pn(z)t"-g(t) is continuous in J t J5 and, g ( t ) and C," Pn(z)tn are analytic in It1 < IpI. Therefore Darboux's method yield

Pn(x) M

r(n+ 1/2)

(1 -

Using nb-"r(a + n)/r(b+ n ) asymptotic formula that

P" 4

1 as n

r(1/2p . 4

00,

we see from the above

P"(2) M (1 - p/a)-1/2p-"/J..n, since r(1/2) = J;;.

(5.9)

22

Case 111. x E (-1,1), so la1 =

IpI, cr # p.

In this case a comparison

function is

d t )= With ,B = e-i6, cr

(1 - t / p ) - ’ / 2 (1 - P / a ) 1 / 2

= eie, so

x

= cose,

+ (1(1--t / a ) - 1 / 2 . Ct!/P)1/2

and we obtain

Therefore einOei9/2

pn(cose) w

e--inOe--iO/2

f i d m

+ fid?%ETe’

or equivalently

In fact (5.10) is uniform in 6 for c 5 0 5 T - E and E E ( 0 , ~ ) . The above example is typical. Darboux’s asymptotic method is useful when we know the singular part of a generating function. Next we discuss Poisson kernels. Let f E L 2 ( E , p ) . Then f will have the orthogonal expansion

c 00

f(x)

N

fnpn(x),

fn

=

/

f(x)pn(z)dp(x),

(5.11)

E

n=O

and { p , } are orthonormal with respect t o p and complete in L 2 ( E ,p ) . The expansion (5.11) is an expansion in L 2 ( E ,p ) . We try, to write an integral representation for the orthogonal series. Let us explore a formal approach, and write 00

0

0

”

It is not clear how to produce f from the above right-hand side unless the integral is S(y - x)f(y)dy, 6 being a Dirac delta function. To remedy

s,

23

this we introduce a damping factor into the series by replacing pn(z)pn(y) by p n ( z ) p n ( y ) t n ,0 < t < 1 and hope for (5.12) with P t ( z , y ) defined in (5.2). The result is that (5.12) holds under some general conditions and the analysis will be greatly simplified if P,(z,y)is nonnegative for z, y E E , and t E ( 0 , l ) . For Hermite polynomials the Poisson kernel is

Formula (5.13) is called the Mehler formula. Clearly Pt(z,y) 2 0 for z, y E R, 0 < t < 1. Indeed (5.12) is the real inversion formula for the GaussWeierstrass transform 63. Poisson kernels are bilinear generating functions. The Poisson kernel for Laguerre polynomials is a multiple of

(5.14)

where F is 00

c

Z"

F(z)= n=O n! ( a +

(5.15)

and is related to the modified Bessel function I,, see 58, ".It is clear that the closed form (5.14), known in the literature as the Hille-Hardy formula, exhibits the positivity of the Poisson kernel for Laguerre polynomials when t E [0, 1). Motivated by the forms (5.13) Sarmanov and Bratoeva 53 considered bilinear series such that (5.16) for c, E R.

Theorem 5.2. (Sarmanov and Bratoeva) If C,"=,c i converges then the condition (5.16) holds f o r all (2, y) E JR x R if and only i f there i s a positive measure p such that

1, 1

c, =

t"dp(t),

n = 0,1,. . . .

(5.17)

24

It is clear that the sequences {cn} characterized by Theorem 5.2 form a convex set whose extreme points are the sequences {t"}, for t E (-1,l); which correspond to measures having a single atom at t. Sarmanov 52 proved the corresponding theorem for the Laguerre series by requiring

(5.18) for all sequences {cn} such that C ~ = o -1. Sarmanov's theorem states that (5.18) holds for all 2 2 0,y 2 0 if and only if c, = tn dp(t) with p a positive measure. Askey gave an informal argument which helps explain these results. Here again the extreme points of the set of sequences {cn} is {rn : 0 5 T < 1) and establishe the positivity of the series side in the Hille-Hardy formula (5.14). The series for the Poisson kernel for Jacobi polynomial is stated on page 102 of Bailey as

Jt

where

(5.20) The representation (5.19) shows the positivity of the Poisson kernel for z,y E [-I, 11,t E [ O , l ) in the full range of the parameters a > -1,P > -1. Bailey also states the kernel

which is positive for z, y E [-1,1],t E [O,l) with a > -1,P > -1 but with the additional constraint a /3 > -1. The question of proving the positivity of Pt(z, y) when the polynomials depend on parameters have been the subject of many research papers, see the monograph by Askey 4 , the treatise by Gasper and Rahman 22 and the references in them.

+

25

Observe that the derivation of (5.9) and (5.10) did not use the orthogonality of P,(x). In fact we started with the recursive definition (5.3)-(5.4) and applied Darboux's method. Indeed (5.9) and (5.10) show that P,(x) is oscillatory when x E (-1,l) and has exponential growth in C \ [-1,1]. The oscillatory nature of P, indicates that the zeros are dense in [-1,1]. Several mathematicians worked on the problem of determining the asymptotic behavior of p,(x) for n large from the qualitative of the recurrence coefficients {a,} and {b,} in (2.2). The model polynomials for this analysis are the Chebyshev polynomials of the first kind {T,(x)},

T,(COS 8) = cos no,

(5.22)

+2

xT,(x) = -Tn+1(Z) 1 --ZL-l(X). 1 2

(5.23)

The normalized weight function is (1 - x')-'/'/n. A sample of results in this direction is the following theorem of Nevai from 46. Theorem 5.3. Assume that b, 4 0, and a, 4 0 in such a way that ca n ( l a , - 1/21 Ibnl) converges. T h e n {p,(x)} are orthonormal with respect to a probability measure p, and p' i s supported o n [-1,1]. The discrete part of p i s finite (may be empty) and lies outside (-1,l). Moreover

+

holds for x = cos 8 for 0 < 8 < n and

'p

depend o n 8 but not o n n.

Note that the normalized weight function for {T,} is (l-x')-'/'/n, and for n > 0, T,(x)} are the orthonormal polynomial. Therefore (5.24) reduces the asymptotics of general {p,(x)} t o the asymptotics of T,(cos 8) p,(x) behaves with 8 shifted by ' p ( 8 ) l n+ o ( l / n ) . In other words as if p , is f i T,. Another theorem of N e ~ a isi ~ the~ following.

{a

+

Theorem 5.4. Assume that Cf"(lu, - 1/21 Ibnl) < 00. T h e n p'(x) i s supported o n [-1,1] and p may have a discrete part outside ( - 1 , l ) . Moreover with x = cos 8,0 < 8 < 7r, the limiting relation

holds.

26

We next derive an integral representation for a general Jacobi polynomial. This representation has the form e n f ( z ) g ( z ) d zso , one can apply the method of steepest descent to this integral and determine the large n asymptotics of the polynomials. The details of the steepest descent method are in Wong’s article in these proceedings, 6 5 . The Jacobi polynomials { P?”’(x)} have the series representation

s,

pp’P’(,)

(-n)k(a+P+n+l)k

=

+n!l ) n

k!(a

The Pfaff-Kummer transformation

+ 1)k

( I1 )--Icn .

(5.26)

is

provided that the series on both sides are convergent. Now apply (5.27) to (5.26) to get

we establish

The use of the binomial theorem and Cauchy’s theorem leads to

P(a’p’(x) n =

&

s , ( 1 + (x

+ 1)2/2)”+”(1+

dz

(x - 1)Z/2)”+PF,

(5.29)

where C is a closed contour so that - 2 ( z f 1)-’ be in the exterior of C . The integral representation (5.29) has a form suitable for the application of the steepest descent method. In fact the integral representation (5.29) can be used to derive the generating function

c

2a-kpR-1

00

P p P )(z)t” =

n=O

+

(1 - t

+ R)a ( 1 + t + R)P ’

(5.30)

where R = dl - 2xt t 2 , 5 8 . A different proof is in 50. One can apply Darboux’s method to the generating function (5.30) and determine the asymptotic behavior of Pp”)(x) for large n and fixed x in different parts

27

of the complex x-plane. The interested reader is strongly encouraged to apply Darboux's method to (5.30), or apply the steepest descent method to (5.29) and establish the following theorem.

Theorem 5.5. Let a and ,8 be real. Then

+ 1)-P/2 [(x + 1)1/2 + (x- 1)'/2] a+P 71+1/2 (5.31) 1 [x + ( 2 1 ) ' / 2 ] , (x2 - 1 y 4

F p P ' ( x ) M (x - 1)-"/2(x X-

1

-

f o r z E C \ [-1,1]. O n the other hand i f 0 < 6 PpP)(coSo) = k ( Q ) cos(N6

< T , then

+ y) + 0 (n-312)

,

d=

(5.32)

where

k(6) = ( ~ i n ( e / 2 ) ) - ~ - ' / ~ ( c o s ( e / 2 ) > - P - ~ / ~ , N = n + ( a + P + 1 ) / 2 , y = -(a

(5.33)

7r

+ 1 / 2 ) -2.

When a = ,8, the generating function (5.30) reduces to a generating function for the ultraspherical polynomials since

(5.34) We made no attempt to cover the application of Riemann-Hilbert techniques to deriving large degree asymptotics of orthogonal polynomials. This is a powerful technique and is covered in Deift's monograph 17. The state of the art of this approach is covered in the recent survey article of Arno Kuijlaars 39. We conclude this section by describing a technique to recover the orthogonality measure(s) p when the recurrence relation (2.2) is given.

Theorem 5.6. Let po = 1, p l = (x - b o ) / a l , and bn E R , n 2 0 , a n > 0,n > 0, and assume that { p n ( x ) } is generated b y x p n ( x ) = an+lPn+l(x)

+ bnpn(x) + a n p n - l ( x ) ,

n > 0.

T h e n there exists a probability measure p such that ~ , p m ( x ) p n ( x )dp(x) = hm,n * Theorem 5.6 gives a converse to the fact that orthogonal polynomials satisfy three term recurrence relations. We shall refer to Theorem 5.6 as the spectral theorem for orthogonal polynomials. Some authors call it Favard's

28

theorem but it was in the literature before Favard’s paper was written. The probability measure p whose existence is guaranteed by Theorem 5.6 may not be unique. We now introduce a second solution {p:(z)} to (2.2) by the initial conditions

and requiring p: satisfies (2.2).

Theorem 5.7. (Markov). Assume that the recursion coefficients {a,} and {b,} are bounded. Then the measure in Theorem 5.4 i s unique, supported o n a compact set E c R and satisfies (5.36)

Moreover, the convergence in (5.36) is uniform o n compact subsets of @\E. Note that if p is unique then (5.36) continues to holds but E = supp(p) may not be bounded. The convergence in (5.36) is still uniform on compact subsets of C \ E . The Perron-Stieltjes inversion formula is

F(z)= if and only if

p ( t ) - p(s) = €O’ flim

J1”

!k@ z-t

(5.37)

F ( u - if) - F ( u + i f ) 2lri

du.

(5.38)

Theorem 5.7 is an instance where the asymptotics of polynomials give crucial information about their orthogonality measure.

6. Applications This section is devoted to some problems which naturally lead to three term recurrence relations and determining the orthogonality measure p or its support sheds some light on the original problem. The first example is birth and death processes. Consider a stationary Markov chain whose state space is the nonnegative integers. Let p,,,(t) be the transition probability to go from state m t o state n in time t. Here

29

p m , n ( t ) does not depend on the initial time. We assume that

pm,n(t)=

{

+ o(t), pnt + o ( t ) , Xmt

1 - (Am

+pm)t +

n=m+l, n=m-1, o ( t ) 7~ = m,

(6.1)

and p m , n ( t ) = o ( t ) ,if Im - n1 > 1. The birth rates Am and death rates p m satisfy po 2 0, p m > 0, m > 0 , and Am > 0 for m 2 0. The Chapman-Kolmogorov equations which describe this process are

+ Pn+lPm,n+l(t) - + p n ) p m , n ( t ) , (6.2) Ijm,n(t) = Xmpm+l,n(t) + pmPm-l,n(t) - (Am + prn)pm,n(t), (6.3) where jl means %. Let us attempt to solve (6.2)-(6.3) by the separation of Ijm,n(t) = L - l P r n , n - l ( t )

(An

variables prn,n(t) = f ( t ) Q m F n .

Therefore f ' ( t ) / f ( t is ) independent o f t , so set it equal t o -x. Now (6.2) indicates that Fn depends on z, so we denote it by Fn(x).jFrom (6.2) we get

-xFn(x) = Xn-lFn-l(x) with

Fo(x)arbitrary,

- (An

+ pn)Fn(x) + p n + l F n + l ( x ) ,

so we take

F - l ( x ) := 0 and Fo(z) = 1. Similarly Q - l ( x )

=0

(6.4)

(6.5)

and QO(z) = 1, and

-xQn(x) = XnQn+l(x) - ( A n

+ ~ n ) Q n ( x+) ~ n Q n - l ( x ) .

(6.6)

Indeed (6.4), (6.6) and the initial conditions imply

Thus we have determined a solution and we multiply by separation constants and sum or integrate over all choices of x. Therefore

30

As t

4

0'1 ~ 1 ~ 2. pnPm,n(t) . . + dm,n. Thus

1

Fm(x)Fn(x)dp(x)= tmbm,n,

(6.9)

so it seems that {F,} are orthogonal with respect t o p. At t 4 co our pm,n(t) cannot have exponential growth, hence x E (0, co),and we obtain

pm,n(t) = 5'm

LW

e-"tFm(z)Fn(x)dp(x).

(6.10)

Observe that (6.10) gives the solution in a factored form and it is not difficult t o analyze the large t behavior of p,,,(t) from (6.10). It turns out the p must be a positive, hence {F,} are orthogonal with respect to p. Furthermore the application of the chain sequence techniques of 97 t o (6.5) and (6.6) show that all zeros of F, and Qn lie in (0, co). The integral representation (6.10) has been established by Karlin and McGregor in 35, 36. Later Karlin and McGregor 38 studied random walks on the state space of the nonnegative integers and defined another sequence of orthogonal polynomials. The random walk polynomials are generated by

+

Ro(x) = 1, R i ( x ) = (1 po/A0)2,

(6.11)

Since the recurrence coefficients in (6.12) are bounded. Theorem 5.7 shows that {R,(x)} are orthogonal with respect to a measure supported on a compact set. Theorem 7.5 proves that all the zeros of R, belong t o (-1, l), for all n. From this fact, one can prove that { R n ( x ) }are orthogonal with respect t o a measure supported on a subset of [-I, 11. The Laguerre polynomials are {F,} polynomials when p, = n,A, = n Q 1. The ultraspherical polynomials are multiples of random walk polynomials with p, = n, A, = n 2v. In fact the Jacobi, Laguerre, Hahn, Meixner, the Charlier polynomials, or there various special cases, are birth and death process polynomials or random walk polynomials. The random walk polynomials corresponding t o

+ +

+

An=cn+p,

p,=n,

p>O,c>O

(6.13)

are very interesting. The case c = 1 is the ultraspherical polynomials. In the case c = 0, the polynomials {R,(x)} are orthogonal with respect to a discrete measure. This measure together with some explicit representations were found in 1958, independently and using completely different

31

techniques, by Carlitz l 4 and, Karlin and McGregor 37. In 1984 Askey and Ismail analyzed the full model (6.13). They proved that the orthogonality measure is absolutely continuous on [-2&/(c + l),2&/(c l)]. When c # 1 it has an infinite discrete part supported in [-1, -2&/(c l)],[2&/(c l),11. The points f 2 & / ( c l), do not support positive masses but are the only limit points of the points supporting positive point masses. Moreover contains explicit and asymptotic formulas for the polynomials and their generating functions. The techniques used axe Markov’s theorem, Darboux’s method, as well as standard special function techniques. Another problem which leads t o orthogonal polynomials is the so called J-Matrix Method. The idea is to start with a Schrodinger operator on R,

+

+

+

+

d2 f V(X). (6.14) dx2 The operator T is densely defined on L2(R) and is symmetric. The idea is to find a complete orthonormal basis in the domain of T such that T the matrix representation of T in { p n ( x ) }is tridiagonal, that is

T

J,-p,TPndx

:= --

= 0,

if

Im - n1 > 1.

We now diagonalize T , that is let T$E = E$E and assume (6.15) n=O

Observe that

E$n(E) = (E$E,p n ) = (T$E,p n ) = (TPn-l$n-l+

Tpn+n

+ Tpn+l$n+lpn, p n ) .

Therefore

+

E $ n ( E ) =$n+l(E)(TPn+l, p n ) $ n ( E ) ( T p n i ~ +n-l(E)(Tpn-l, ~

+

n

)

n ) .

(6.16)

If ( T p n , p n * l ) # 0, then (6.16) is a recurrence relation for a sequence of polynomials { & ( E ) } . It is easy t o see that (6.16) can be reduced to (2.2) if and only if

(TP, Pn-l)(Tpn-l, p n ) > 0. This is indeed the case since (Tv,,pn-l) = ( p n , T p n - l ) = ( T p n - l , p n ) . Heller, Reinhardt and Yamani 24 introduced this method and applied it to

32

physical problems. In particular they identified the orthogonal polynomials { & ( E ) } which arise in the harmonic oscillator, the Morse oscillator, and the hydrogen like atom. The polynomials {qn(E)}are Meixner polynomials in the harmonic oscillator model. The hydrogen atom with a Coulomb potential was more challenging. In the case when the Coulomb potential is repulsive, i.e. the electron and nucleus have charges of the same sign, & ( E ) are the polynomials of Szegii and Pollaczek, see 5 8 . In the physical case when the Coulomb potential is attractive, the & ( E ) are the AskeyIsmail polynomials generated by (6.12)-(6.13).

7. Zeros of Orthogonal Polynomials and Eigenvalues Let A , be the infinite Jacobi matrix

[T;!:), bo

A,=

a1

0 0 ...

...

and let AN be the N x N truncation, that is delete from A, the columns N 1,N 2 , . . . , and rows N 1, N 2 , . . . , and AN is what is left.

+

+

+

+

Theorem 7.1. W e have

& ( A ) = det(XI - A N ) ,

(7.1)

where “det” stands for (‘deteminant)’. In other words PN(X)i s the characteristic polynomial of A N . Proof. Let Q N ( X )= det(XI - A N ) . Verify that Pl(X) = Ql(X),Qz(X)= PZ(X). For N > 1, expand Q N + ~ ( X about ) the last row then expand the cofactor of ahrX - b N about its last column. The result is

QN+I(X)

= (A - ~ N ) Q N ( X ) - ~

%QN-~(X),

which is the same recurrence relation satisfied by Pn(X),see (2.6).

0

Theorem 7.1 relates zeros of orthogonal polynomials to eigenvalues of tridiagonal or Jacobi matrices, where linear algebra techniques are available to study the locations of eigenvalues, Horn and Johnson 27. Some of the questions in this area of research are:

1. Find approximations to the zeros of P,(z).

33

Let

be the zeros of Pn(x). Find the large n asymptotics for z n , k and x,,,-k for fixed k. If pn(x)depends on a parameter, what happens to xn,k as the parameter increases (decreases). Find inequalities and bounds for the zeros of Pn(x). Ask the same questions for zeros of special functions. Let us address first the question of monotonicity of zeros of a transcendental function by turning the problem t o monotonicity of eigenvalues of a symmetric eigenvalue problem. Consider the Bessel function

(7.3) It satisfies the differential equation 2

x y Replace x by

&i x

It

+ xy‘ + ( x 2- u2)y = 0.

(7.4)

and write (7.4) as

I d x dx

--- ( x $ )

+ -y5U 22

= xy.

(7.5)

For u > 0 consider the eigenvalue problem consisting of (7.5) and the boundary conditions y(0) = 0,

y(1) = 0.

(7.6)

For u > 0, the Bessel equation has one unbounded solution (= Y,,(z))and J,,(x) is the bounded solution. Hence the solution of (7.5)-(7.6) must be y(z) = cJv(&i x) and A > 0 is such that Jv(&i) = 0. The function J,,(x) may have a trivial zero at x = 0 but for u > -1, it has only real and simple zeros which are symmetric about x = 0. Let 0

< j”,l

jv,2

< ...,

be the positive zeros of J,,(x). The question we wish to address is how does change with u when u > 0. To address this question let us consider a .symmetric operator T,, depending on a parameter u and is densely defined on L2(wdx,E ) , where ju,k

34

The symmetry of T, means (T,z,y) = (z,T,y) whenever z and y belong to the domain of T,. Let T,y = X(v)y. We now give a heuristic argument t o prove the Hellmann-Feynman theorem:

SE ~ ( z > [ T , y ( ~ ) ] y ( z ) d ~ , ~

To prove (7.8) differentiate the relationship

X(u) =

t o get

from which (7.8) follows. A similar argument shows that (7.8) will continue t o hold if the inner product (7.7) is replaced by ca

n=O provided that w, 2 0, f = { f n } , g = {gn}.

3

Exercise: Find a rigorous justification of (7.8) by defining in a suitable way. The difficulty is in justifying the differentiability of X(Y) and interchanging differentiating and integration. The Hellmann-Feynman theorem was introduced in 2 5 , 21. For recent works see 34, 32. In particular 32 contains applications to q-Bessel functions and certain classes of orthogonal polynomials which do not satisfy differential equations of Sturm-Liouville type, or Schrodinger type (6.14). It is important to note that the Hellmann-Feynman theorem uses only the eigenvectors (eigenfunctions) and not their derivatives. The derivatives of the eigenfunctions are usually complicated. Example 1. Consider the Bessel operator I d U2 T,y :=

---z (zg) + 2y,

u

> 0.

(7.10)

35

The eigenvalue problem consisting of Tvy = Xy together with the boundary conditions (7.6) is symmetric with respect t o the inner product (7.7) with E = [0, l],w(x) = x. Let X = jy2,k. Thus (7.8) gives (7.11) The right-hand side of (7.11) is clearly positive, hence v for v > 0.

j , k

increases with

Example 2. The Bessel functions satisfy the three term recurrence relation (7.12) Let fo(x,v) = 0,

fn(x, v) =

Jv+n(x),

72 = L 2 , .

. . v > -1. (7.13) 7

With X = l / j v , k , (7.12)-(7.13) give

n = 1 , 2 , . . . ,v > -1. Clearly (7.3) and (7.13) imply C y f i ( j v , k , v) < cm, thusc(fl,fz, ...)TisaneigenvectorofA, witha, = 2 - ' ( ( v + n ) ( v + n 1))-1/2, and bn = 0. Here c is a constant to make the eigenvector have norm = 1. Thus (7.8) implies

The right-hand side is

The second sum is a telescoping sum whose value is zero. In the first sum apply (7.14) and evaluate c t o obtain

36

Theorem 7.2. W e have (7.16)

(7.17)

+

+

Proof. Since v n 3 v 1, we see that the right-hand side of (7.15) is majorized by j v , k / ( v 1) and (7.16) follows. Integrate (7.16) between 1/2 and v and use j l / z , k = k7r to derive the first formula in (7.17). The second formula in (7.17) follows by integration and using j - 1 / 2 , k = ( k - l/2)7r. 0

+

3

Theorem 7.3. If is positive semi-definite (definite) then the eigenvalues of T, are increasing (strictly increasing) functions of v. Proof. Apply the Hellmann-Feynman theorem.

0

We conclude the material on the Hellmann-Feynman theorem by giving a proof of a theorem of Loewner 4 1 using Theorem 7.3. Loewner’s theorem asserts that if A and B are n x n Hermitian matrices and B - A is positive definite, then Xk(B) > Xk(A), where X1(A) 2 X2(A) 2 ... 2 X,(A) are the eigenvalues of A , and similarly for B . To prove this introduce a vdependent matrices C ( v ) C , ( v )= vB (1 - v)A. Clearly = B - A, and Theorem 7.3 establishes > 0, since B - A is positive definite. The theorem follows because C(1) = B,C(O) = A. If the positive definiteness of B - A is replaced by positive semidefiniteness then we conclude that & ( B ) 2 Xk(A). Another proof is in 27. In the rest of this section we introduce and apply chain sequences to locate intervals containing all zeros of orthogonal polynomials. Chain sequences were first introduced by Marion Wetzel in her doctoral dissertation and she, jointly with Wall, applied them to continued fraction, 61, 6 2 . It was Ted Chihara, however, who popularized chain sequences among researchers in orthogonal polynomials through his papers, lectures and his influential monograph 1 6 .

+

Definition 7.1. A sequence { ~ } isya chain sequence if there is a parameter sequence {g,}r such that 0 5 go < 1 , 0 < gn < l for n > 0, and cn = gn(l - gn-l),n > 0. If {cn}y is a finite sequence we call it a finite chain sequence. The sequence {g,} is called a parameter sequence for { c n } .

37

i}.

It is clear that g, = 1 / 2 is a parameter sequence for { The parameter is also a parameter sequence however, may not be unique. Indeed { &} sequence for {

i}.

Theorem 7.4. (Wall-Wetzel). The Jacobi matrix AN i s positive definite i f and only i f (i) and (ii), (i) (ii)

b, > 0,O 5 n < N , {ai/b,b,-l :1 5n

< N } is a chain sequence,

hold. The proof consists of performing row operations t o reduce AN t o an upper triangular matrix. The positive definiteness is equivalent to the positivity of the diagonal elements (pivots) in the reduced upper triangular matrix. If AN is positive definite then (i) holds and g, can be defined recursively by go = 0, then prove that a f / [ b k b k - l ( l - gk)] E (0, l ) ,so we set gk+l = a i / [ b k b k - l ( l - g k ) ] . The converse can be easily verified.

Exercise. Write down the details in the above proof, or read it in Another proof is in l6

33.

Theorem 7.5. A sequence {c,} i s a chain sequence zf 0 < cn 5 d,, n > 0 , and {d,} is a chain sequence. Theorem 7.6. All the eigenvalues of AN belong to ( a ,b) i f and only if the following two conditions hold: (i) (ii)

a < b, < b a i / [ ( x - b,-l)(x - b,)], n x = a and at x = b.

=

1 , . . . , forms a chain sequence at

Proof. Clearly all the eigenvalues of AN belong to ( a , b) if and only if AN - a 1 and bI - A are positive definite. The theorem then follows from Theorem 7.3.

0

Example. The Hermite polynomials { H , ( x ) } satisfy 2 x H n ( x ) = H,+l(X)

+ 2nHn-1(x).

The orthonormal polynomials are 2 - " / ' H n ( x ) / f l and w(x) = e - x 2 / & , x E R. The zeros are symmetric around x = 0. Moreover

38

b,

= O,a, =

m.Let ( N - 1)/2 A2

5

1 4,

A > 0.

(7.18)

Therefore {aK/A2},n = 1 , . . . ,N - 1 is a finite chain sequence, since {1/4} is a chain sequence, and Theorem 7.3 implies that all the zeros of H N ( ~ ) lie in (-A, A), that is (Recall that the Chebyshev polynomials of the second kind {U,(x)},

d m ,d m ) .

U,(x)= sin((n + l ) O ) / sin 8, x = cos 8.

(7.19)

They satisfy

1 1 2 2 The largest and smallest zeros of U N ( ~are ) fcos[n/(N

xU,(x)

=

1 4cos2[n/(N

+

-Un+l(x) -Un-l(x).

+ I)] '

(7.20)

+ l)].Therefore

n = 1,... , N - 1,

is a chain sequence for every E > 0. After replacing 1/4 in (7.18) by the above finite chain sequence we see that at the zeros of H N ( x ) lie in [-AN A N ]I

AN = cos (2) d m N+1

Ismail and Li

30

.

used Theorems 7.3 and 7.4 to prove the following.

Theorem 7.7. Let {C,}F-' be a finite chain sequence and let x,, yn be the roots of (x - b,)(x - b,_l)C, = a:, and x, 2 y,. S e t

B

= max{x, : 0

< n < N } , A = min{y,

:0

27r.) We will denote by O ( V ) the @-algebra of holomorphic scalar functions on V . 2We insist on the fact that with this definition Poincar6 missed completely the subtle phenomenon of the summation at the smallest term.

50

Fig 1.1.2

Definition 1.1. Let f E O(V) (holomorphic on the open sector V). Let +m

f E C anz" E @[[z]].We will say that f is asymptotic to f on V (in the n=O

classical or Poincark's sense) if, for every strict3 subsector W 4 V, there exists some positive constants 1Mw,"(n E N) such that, for every z E W , the following estimates hold:

I

p=o

I

-f

We will denote f E d ( V ) and f on V. The set d ( V ) of functions in O(V) admitting an asymptotic expansion is clearly a sub-@-algebra of O(V). Moreover, if f E d ( V ) admits E @[[z]]as an asymptotic expansion, then its derivative f' admits f^' as an asymptotic expansion. For elementary properties of asymptotic expansions, we refer the reader to W.Wasow's book [33].

f

The asymptotic expansion, when it exists is clearly unique. The Taylor map J : d ( v ) + @[[z]],f H is an homomorphism of differential @algebras. The basic idea of H. Poincark was that if f E d ( v ) admits E @[[XI] as an asymptotic expansion, then in some sense, we can consider f as a sum of f . Unfortunately such a sum is not unique. The non-uniqueness corresponds to the "Error Space" kerJ = d"O(V), which is the differential ideal of d ( V ) of the infinitely flat holomorphic functions on V.

f

Example 1.1. V = {Rex > 0}, f(z)= exp-S, f 3A subsector W C V is strict if

f

N

0.

w - (0) c V (wbeing the closure of W in C).

51

We insist on the derivation property. It works because we deal with holomorphic functions and because in the definition of asymptotics we asked uniqueness only on strict subsectors. The proof follows immediately from the following lemma.

Lemma 1.1. Let g E O(V). IJ for every strict subsectors W 4 V, we have estimates Ig(x)l < Cw(xln+’, then we have similar estimates o n the derivative: Ig’(x)l < CbIxln. Idea of the proof. Let W’ 4 W” 4 V. We denote by D ( x , d ) the closed disc (in C) with center x and radius d > 0. There exists b > 0 such that, for every x E W’, we have D(x,blxl) c W”. Then we can write a Cauchy integral

and our lemma follows. It is easy t o prove that f E d ( V ) if and only if, for any have

Then f

- f^

=

W

+ V, we

+== C a,P. n=O

We have seen that kerJ = d 0), when z -+ 0, lv(x)l5 Ke-* ( K , a > 0 independent of z). Considering these three observations, it is natural to try to replace Poincark Asymptotics by a new asymptotic theory explaining 1, 2, 3. The good news is that such a theory exists: it is Gevrey Asymptotics. In fact “more or less”:

1

2

3

we are in the case of Gevrey Asymptotics.

Gevrey Asymptotics were discovered by G. Watson at the beginning of the XXth century. But unfortunately it meet more or less no success and was forgotten. I rediscovered it (and gave it its name ..., in relation with M. Gevrey work on partial differential equation ) at the end of the ~ O ’ S , and developed it systematically in relation with the applications [24]. G. Watson’s work was rejected because mathematicians was thinking that its field of applications was extremely narrow. (G. Watson applied his theory only to some special functions: r-function, Bessel functions ... ). In fact, as I will explain later, its field of application is today extremely large, containing whole families of analytic functional equations (ordinary differential equations: without restrictions, singular perturbations of ordinary differential equations, some problems of partial differential equations ... ). If G. Watson’s work was forgotten for a long time, it is worth t o notice that there is however a “red thread” going from G. Watson to S. Mandelbrojt (though some works of R. Nevanlinna, Carleman and Denjoy). Gevrey asymptotics is an essential step towards exact asymptotics. Moreover it is exactly the good asymptotics for singular perturbations and it allows us to understand phenomena like delay in bifurcations, ducks phenomena, Ackerberg-O’Malley resonance, ... or perturbations of Hamiltonian systems (adiabatic invariants, Nekhorosev estimates, ...)

I will begin with my favorite example (Euler series):

n=O

It was introduced by L. Euler in his paper ‘LDeseriebus divergentibus”. His

55

aim was to "compute" numerically the infinite sum +m , --

j(1) =

C(-l)"n! n=O

In relation with this problem, Euler considered the linear differential equation x'y' y = x. Solving it by the "variation of constants", we get:

+

and (setting

$ =5)

+Ceh.

-dt We remark that the integral

is the unique solution of our differential equation which is bounded on R+. Euler concluded from this remark that we can consider f(x) as the sum of the power series expansion f(z) (which is a formal solution of the equation). Finally he got C (-1)".n! = f"(l)'. n20

We will explain why the idea of Euler is reasonable (x > 0). We set

+ ... + (-l)n-l(n

f n ( z )= 2 - 1!x2 and

1

+m

Rn(2)= (-1)n

@[I.

tne-; l+tdt.

+

We have f(x) = f n ( x ) Rn(z)for every n E

IRn(x))
0 be fixed. Then f 2 p + 1 ( z ) - f2,(z) = (2p)!z2Y+lfirst decrease, then growth when p growth. The smallest difference gives the best approximation for f(z). This corresponds to the summation at the smallest term. We have the following estimates: n!xn+l = nx ( n - l)!zn

and the smallest term is reached at N

M

$, N

=

[$I.

From Stirling formula:

Therefore, when x 5 0, the equality of our approximation is exponentially good. We can understand with this example why divergent series are better than convergent series for numerical applications. The case of logr is similar (A. Cauchy). We will discuss a striking example due to Stokes.

~

light rays

Fig 1.2.1 We start from a caustic in classical optics. The caustic is wrapping the light rays and separating lighted zone and darkness zone. If we consider caustics in wave optics, then we can observe interference fringes. The problem considered by Stokes was to compute the distance between the fringes using theoretical ways and to compare with experiments. (Physicists measured 30 fringes with an accuracy of 4 digits.) The intensity of the light along a small piece of line transversal to the caustic has been computed by Airy. With a good choice of unities it is ( A ~ ( z )where ) ~ , A Z ( Z )is~Airy function ( z being a parameter transversally to the caustic). We have:

1IT

~ i (= ~ )

fa

1 cos(-t3 3

+ 3t)dt

57

and this function is a solution of the Airy's equation y" - zy

= 0.

Airy function admits a convergent asymptotic expansion a t the origin z = 0:

Airy tried t o compute the fringes using this convergent expansion. He could only get one fringe (with 4 digits). Conversely t o Airy, Stokes used expansions at infinity ( z = 00). Using stationary phases method, we get the expansion

The power series appearing in this expansion is clearly divergent. However, using this expansion and summation a t the smallest term (in fact an improved version), Stokes was able t o compute all the 30 fringes with an accuracy of 4 digits, compared with physical experiments. (He failed only for the first fringe: he got only 3 digits...). We can see on this example the apparition of Gevrey estimates and exponentially small functions. Returning back t o this example several years later, Stokes discovered what is called today Stokes Phenomenon (March 17, 1857, three a.m.; cf. ~41). We will give now some precise definitions.

Definition 1.2. Let V be an open sector of C (with its vertex at the origin). Let f E O ( V ) . Let f^ E @[[XI] and s E R. We will say that f is asymptotic t o f^ in Gevrey order s sense, if, for every strict subsector W 4 V, there exists positive constants CW > 0 and AW > 0 such that, for every n E N, z E W : n-1

1~1-"lf(z) -

C aPzPI5 c ~ A b ( n ! ) ~ .

p=o

f^ E C[[z]lsand f E d,(V). Remark 1.1. If s = 0, then f E Cz: f^ is convergent and conversely. If s < 0, we set s = - i ( k > 0). Then f E C[[z]lsif and only i f f is an We will denote 0

0

entire function with an exponential growth order 5 k a t

00.

58

By definition, we will set @[[z]lm = @[[XI]. Definition 1.3. I f f =

C anzn satisfies n20

lan[ < CA"(n!)"

for some positive constants C, A > 0 , then we will say that order s and we will denote f^ E @[[XI],.

f

is Gevrey of

It is ideal that the image of d,(V) by J is contained in @[[z]],.We will consider the map A,(V)

@[[zlIs.

We have the following result [24], [l],[2], [31]. Proposition 1.1. (i) @[[z]],is a sub-differential algebra of @[[z]]. (ii) d,(V) is a sub-differential algebra of d ( V ) .

It is natural t o study the subjectivity of the map J : d,(V) -, @[[x]],. The answer depends on the opening of the sector V. It is yes for narrow sector (ie. if opening of V < f ; Ic = and no for large sectors (opening of v > f ) .

t)

Theorem 1.2. (Borel - Ritt Theorem, due to Ramis) If V is a narrow sector, that is the opening of V is strictly smallest than f = T S , then the @[[z]],is surjective. map J : d,(V) We will give an idea of a proof (following an idea of B. Malgrange). We will limit ourselves t o the case Ic = s = 1. We will use an incomplete Laplace transform. Let

f

+m

a n 9 . By definition its formal Borel transform is the power

= n=l

series

Sf=C-

an+l

-

n=O

(n

+ l)!t".

Then f E @ [ [ z ] ] ~ Sf E @{t}(i.e. convergent). Formally, the Laplace transform of a function t H p ( t ) is the function +m

z

H

cp(t)e-$dt.

59

Here, we denote by cp(t)the sum of Bf = @(t) .Unfortunately, in general we cannot extend 'p analytically along Rf.Moreover, even if this extension exists, in general the Laplace integral does not converge ... But, if R > 0 is the radius of convergence of @ (in the t-plane), then the integral

will exist for 0 < r < R, z E C,and will define a function of z asymptotic to on V in Gevrey 1 sense. > 0, we can adapt the proof. We replace For other values of s = the incomplete Laplace transform by an incomplete k-Laplace transform. Formally we set

f

p k f (z) =

f(.'),

Lk = P i 'Lpk,

Bk = PL'Bpk.

We denote AS-'"(V)the kernel of the map J : A , ( V ) + @[[z]],.

Heuristics. With narrow sectors, Gevrey asymptotics is very similar to Poincare's asymptotics. It is in some sense a smooth theory (typically, we will have existence theorem, but non unicity). With large sectors, Gevrey asymptotics appears as a completely new asymptotic theory. We get exact asymptotics. It is a rigid theory (typically existence theorem are exceptional, but we have unicity like for elementary Cauchy theory of holomorphic functions ). For large sectors, the following lemma is crucial. Lemma 1.2. ( W a t s o n L e m m a ) Let k > 0 . If the opening of the sector V > i and,if f E A'-i(V), then f is identically zero o n V . This lemma is a simple consequence of Phragmen-Lindelof maximum principle. It follows from Watson lemma that, for large sectors, the map J : A,(V) @[[XI], is injective. As we will see later it is no longer surjective: --f

04 A,(V)

@.[[z]],.

The power series belonging to the image of J are very special (this is related to the notion of k-summability, for k = l/s, cf. next paragraph). For narrow sectors, the situation is in some sense the opposite. We have a short exact sequence (s = l / k ) :

0 ---f A'-k ( V )

+

U V ) 5 @[[z1ls.

60

Definition 1.4. Let k > 0. We will say that f E O(V) admits an exponential decay of order k on V (when 2 -+ 0), if, for every strict subsector W + V, there exist positive constants Kw > 0 and aW > 0 such that

I f(.)I for

2

5 KWe-awllzlk

E W.

Proposition 1.2. The following conditions are equivalent: (i)f E d 0. Let f^ E such that there exists an open sector V whose opening > i and a holomorphic function f E d + ( V ) such that f is asymptotic to f on V in Gevrey sense. We will say that is k-summable in the direction d (d being the bisecting line of V).

f^

Then we will say that f is the k-sum of f^ in the direction d (d being the bisecting line of the sector V). In this case we remark that f is unique (up t o opening and the radius of the sector V , d remaining fixed). Moreover, we must have f^ E

@[XI];.

Notation.

f EC { X } ~ , ~ .

We will see that the inclusion @ { x } $ ,c~ C [ [ x ] ]is; strict. Of course, if

f^

is convergent, f E C{z}, then f^ is k-summable in the direction 4 (any k > 0 and any d ) and the classical sum and k-sum coincide on their respective domain of definition : @{.} c (C{z};,d.

Remark 1.2. This summability definition sounds quite abstract. In fact it is extremely useful: 0 With it, it is easy t o prove the elementary properties of summability (sum, products, differentiation ...). 0 It is easy t o prove in many cases issued from dynamical systems problems that some formal power series solution f^ is k-summable, looking a t the underlying “geometry” of the system. (Ramis-Sibuya theorem is an efficient tool ; cf. below). 0 What is apparently bad with our definition is that it is seems a priori impossible t o compute the sum. With other definitions of summability (by

61

E. Borel and his followers, like Leroy), conversely we have explicit formulae for the sums but summability seems extremely difficult to check8 The good news is that the two definitions are in fact equivalent. This is very convenient because we can choose one or the other following the applications we have in mind.

Theorem 1.3. (Ramis) For f^ E @[[.I], k-summability in the direction d is equivalent to Borel-Leroy-Nevanlinna summability in the direction d .

Proof. cf. [14], [l]. We recall the principle of Borel-Leroy-Nevanlinna summation. We will limit ourselves to the case of k = 1 (Borelg) It is easy to derive the general case (Leroy, R. Nevanlinna, using an operator p k ) . We choose also d = R+ for simplicity. The principle is the following. We start from a power series expansion: +a

f^ = C an.",

we associate to

f^

a new power series expansion (in the t

n=l

variable): +W

n=O

We suppose that: a) $3 is convergent (jE ~ [ [ z l l l ) ; b) the sum 'p of $3 in an open disc centered at t = 0 can be analytically extended in an open sectorial neighborhood of R+;

Fig 1.3.1 sThere was a strong prejudice against Borel-summation due to this fact (Mittag-Leffler, 1900)

91n fact Borel summation is a little more general than 1-summation.

62

c) the integral

1

+W

j(x) = ~ c p ( x=)

cp(t)e-?dt

converges for x E V (the sector V being bisected by R+ with opening > 7r = f). Then, by definition f(x) = Lcp(x) is the Borel-sum of f^ in the direction R+. In the next paragraph, we will recall basic facts about Laplace transform. Using the first (abstract) definition of k-summability, it is evident to check the following.

Proposition 1.3. Let k > 0. Let d a fixed direction. The set @ { x } * ,is~ a C-sub-digerential algebra of C [ [ x ] ] ; . Definition 1.6. If f E @[[XI] is k-summable in all the directions but perhaps a finite set, we will say that it is k-summable and will denote E @{x}*.

f

Proposition 1.4. (i) Iff is k-summable in all the directions, then convergent (and conversely). (ii) @{x}; is a sub-diflerential algebra of

f

is

@[[XI]*.

Remark 1.3. The inclusion @ { x } l ; d

c @[[z]]l is strict.

Bf

Indeed, consider f E @ [ [ x ] ]such 1 that = @ admits a radius of convergence 1 with the circle of center 0 and radius one as a natural cut. Then it is not possible to extend cp analytically along d and @ @ { x } l , d . A similar argument can be used to prove that the inclusions @{z};,~ c @[[XI]; are all strict.

f

1.4. Laplace transform

Fourier transform. We will use real conjugated variables t and 0. Then the integral

l+w

‘p(t)e-tzdt(= .F[’p(t)e-atIp+](z))

+

is convergent for z E C, Re z > B . Here, we set z = u it and we denote by IR+ the characteristic function of Rf c R. We suppose that we have for ‘p, ‘p’, ‘p” estimates of type (1.4.1), and moreover: ‘p(0) = ~ ’ ( 0=) ‘p”(0) = 0. Then using Fourier inversion formula, we get 1 a+im

v(t)=

/

L‘p(z)eztdz.

a-am

By direct computations:

1 L(1) = -, 2

1

L ( t )= 22

and

-dz

etz

= Res(0, -)

22

=t

(u

> 0);

L(t2 ) = -,2

23

The Laplace transform in the complex plane Proposition 1.5. The Laplace transform C ; ‘p --f g is an isomorphism between two spaces: (i) Holomorphic functions ‘p o n an unprecise (narrow) sector C = {IArgtl < E } ( E > 0 ) with an exponential growth of order one and finite type at infinity in this sector: I‘p(t)l5

AeBltl

and admitting an asymptotic expansion at 0: m

64

(ii) Holomorphic functions g in an unprecise sector

and admitting an asymptotic expansion at

00

in this sector:

Moreover we have an+l = n! b,. We have (cf. Fig. 1.4.1):

c+im

Fig 1.4.1 From complex Laplace transform, we get the Borel-Laplace formalism. We use the conformal transformation z = $ (automorphism of C* = C ( 0 ) ) . We set g ( z ) = f ( x ) . We have (cf. Fig. 1.4.3):

65

I

Fig 1.4.2

Fig 1.4.3 By definition, B is the (actual) Borel-transform. Laplace transform, the transform

We will also call

The formal counterparts are: tn-1

B(x") = _____ ( n- I)!'

E(tn)= n!xnf1

and

g(1)= 6 (Dirac at t = 0),

c(6)= 1.

1.5. Ramis-Sibuya theorem and k-summability Theorem 1.4. W e assume that {V1,..,,Vm) is an open covering of the punctured disc D* = { x E @,O < 1x1 < r } by open sectors, such that

66

the three by three intersections are void. Let f1, ...,f m be a collection of holomorphic functions satisfying the following conditions: (i) each f j E O(V,) is bounded o n V, (j= 1,2, ...,m); (ii) fj,j+l = f j + l - f j E A 0. If (ii) is replaced by (ii)’ f j , j + l E As-k(Y,j+i)JA then f^ E @[[XI]; and f j f in Gevrey sense o n V, (j = 1,2, ...,m). N

Corollary 1.1. If f j E A 0 ) by open sectors. ( W e suppose that 3 by 3 intersections are void.) Let cpj,j+l E As-’((V,,j+l) (‘h 1 = 1”) be a given family of functions ( j = 1 , 2 , ..., m ) . Then (reducing R > 0 i f necessary) there exists a family of functions {cpj}j=1,2,...,m , such that: (i) ‘pj E A(V,) (j = 1,2, ...,m). (ii) cpj+l - cpj = cpj,j+l (j = 1,2,..., m; “m 1 = 1”).

+

+

Moreover, if cpj,j+l E AS-k(V,,j+l) (j = 1 , 2 , ..., m), then (i)”pj E A$(%) ( j = 1,2,...,m).

Proof. By linearity, we can suppose that cpj,j+l = 0 except for j = 1, j 1 = 2. Then we set ’ p l , ~= g, and V1,2 = Vl n V2; V = {x E @ , a< argx < ,B, 1x1 < R}. We have ,B < a 27r. We will consider “ramified sectors” (using the Riemann surface of the Logarithm). In particular we introduce the ramified sector = {x E a r g x E [a,,B 27r]}. (cf. Fig 1.5.1 ) We choose 0 < r < R and denote y = [0,7ei6] the corresponding segment, for a fixed direction of argument S E [(.,PI. For a < 6 < 6’ < p,

+

+

o*,

+

67

Fig 1.5.1 we set y’ = [0,7ei6‘]. Let p’ be the circle arc joining reib to Tei6’. We introduce

and h,!, h,! by analog formulae.

Fig 1.5.2

By Cauchy integral formula, we get

h,, - h,, - h,

0

for z not belonging to the “triangular” domain delimited by y,p’, 7’. The function h,(z) is defined on C - y,in particular for arg 5 E (y, y 27r). For z such that arg z E (y,y 27r), we can replace the contour of integration y

+

+

68

by pl

+ yl, without changing h,(x):

+

hy(x) = hp‘(X) h,’(X).

’

+

But h,, (x) h,! (x) is defined and holomorphic for x such that 1x1 < T and argx E (y1,y’+27r). Therefore we get an analytic extension of hy(x) to the ramified sector {x E d* : 1x1 < T , argx E (6,s’ + 27r)). Moving 6’ and therefore yl towards argx = p, we get an analytic extension to the ramified sector (1x1 < T , argx E (6,p 2 ~ ) ) .Then we can suppose a < 6’’ < 6 < p and move 6” towards a. By a similar argument, we get an analytic extension of h, to the ramified sector: { 1x1 < T , arg 2 E ( a ,p 2.)). We denote h this extension. Let x be such that a < Algx < P. We choose b’, 6” such that a < S” < arg x < S’
0). We write

+

1

-=-

therefore

We set

n- 1

xp

xn

1

69

and we have

Because I a r g z - 61 2 E , we have (tei6- 21 2 1x1 sin&,therefore

The function g being infinitely flat at 0, we have

therefore

It follows that

If, moreover, g E d 0), if it is k-summable in every direction d but perhaps a finite number (the so-called singular directions o f f : d l , ...de). The finite set C(f) = {dl, de} is the singular support of C(f) = 0 H E @{a:}, i.e. convergent. It’s easy t o see that, when d moves between two consecutive singular directions di and di+l, then the sums fd o f f in the direction d give together analytically and define a holomorphic function fi on a “sector” %lo (cf. Fig 1.5.3):

f;

f

7r

arga: E (argdi - -,argdi+l+ 2k

7r

-). 2k

Fig 1.5.3 “it is an eye-shaped sector, the radius 1x1 can tend to zero when a r g s & ; it can be ramified. arg di+l

+

+ arg di -

& or

71

The functions fi and fi+l, ( i = 1,...,i? - 1) have the same asymptotic expansion at the origin: f i , fi+l f”, in Gevreyi sense. Therefore fi+l fi = fi,i+l is asymptotic to zero on V,,i+l = V , fl V,+l in Gevrey $ sense (i.e. is exponentially decreasing with an order k). The eye-shaped “sector” &+I admits an opening exactly equal to i. N

There is a converse to this property which ( even if it seems surprising) will give a very efficient way to prove that a power series derived from a problem of dynamical system is k-summable.

Theorem 1.5. W e assume that {Vl, ..., V}, is an open covering of the punctured disc D* by open sector, such that the 3 by 3 intersections are void. Let { f1, ..., f,} be a collection of holomorphic functions satisfying the following conditions: (a) each f j E O(V,) is bounded o n V, ( j = 1,..., m); (ii) fj,j+l defined in V,,j+l = V, n & + I can be extended in an holomorphic function defined o n an eyeshaped ((sector” with opening 2 such that this extension (denoted also fj,j+l) satisfies fj,j+l E A- 0) and two values f + ,f - on {Rex < 0). The “branches” f + and f correspond respectively to fdt and fd- where d+ = eie+R+,d- = eie- R+, B+ = T a , 0- = T - a, where a > 0 is “small”. We can easily compare f + and f-:

+

We can LLdeform” y in a bounded contour y‘ without changing the integral (Cauchy theorem and elementary estimates). Then we get dt

= 2i7r

e-t/x Res(-; t = -1) l+t

= 2he1/”

73

So, f-(x) - f + ( x ) = 2i7re1/" which is,.as we waited, exponentially small on the half-plane {Rex < 0) : f - - f + E dS-l({Rex < 0 ) ) . The functions f,f , f +, f - are actual solutions of the linear non homogeneous differential equation (Euler equation):

2 y l + y = x. The function f-(x) - f + ( x ) = 2i7re1/" is an actual solution of the corresponding homogeneous differential equation:

x2yl+ y = 0. We can now describe the monodromy of the problem, that is what happens t o the solutions when we turn around the origin by analytic continuation. We start from f on {Rex > 0) and we extend it by analytic continuation along a loop turning around 0 in the positive sense (we replace R+ = eioR+ by d = e i e R f with 8 ~ ] O , 7 r [ ) ;firstly we get f-. When d crosses R- (8 crosses 7 r ) , f- jumps to f + = f - - 2ire'l". Therefore in this region f + 2i7re1/" is the analytic continuation of f . It works for Of E] - 7 r , O [ . Then we get an extremely important phenomena (an example of the Stokes phenomena): near the direction R- (the singular direction of f : C(f) = {It-}), the two functions f f and f- admit the same asymptotics (i.e. f); they differ by 2ine1/" which is infinitely flat. The function f f 2i7re1/" admits the asymptotics f on {Rex < 0), but when we cross the line -iR+ = ePiT/'R+, f f keeps the same Asymptotics f and 2i7re1lX jumps, turning its asymptotics (i.e. 0), and becomes infinitely big (exponentially big) on {Rex > 0 ) . On the line -iR+ it is bounded and oscillates. The line -iRf is (I think improperly12) named Stokes line by many authors. Turning backwards we get a similar phenomena on the other Stokes line: iR+ = ei.rr/ZR+. After one turn around the origin (in the positive sense) by analytic continuation, the function f is replaced by the function f 2i7re1/" (it is an a f i n e transformation in the affine space of solutions of Euler ordinary differential equation). We observe that f 2i7re1/" is not asymptotic t o Its dominant part is 2i7re1/" which exposes when x -+0 in the half plane {Rex > 0 ) .

+

+

+

+

f.

I2The true Stokes phenomena happens when we cross the singular line R-. At the beginning we cannot notice it by asymptotics and we observe it only when we cross

-iE%+.

74

Airy equation (cf. [17]) This second example is the description of the solutions of Airy equation

y“ - zy = 0

(1)

near z = 00, using asymptotics. Stokes discovered Stokes phenomena studying these solutions. We consider Airy ordinary differential equation on the Riemann sphere P’(C) = C u {co}. Its only singular point on this Riemann sphere is clearly z = co. A first consequence is that any solution on a small disc centered at zo E C extends analytically to all C: it is uniform (single valued) and if we turn around co along a loop by analytic continuation, we will go back to the same solution: the (actual) monodromy transformation is the identity. Airy ordinary differential equation (1.5.1) admits a basis of convergent power series solutions at z = 0 (cf. 1.2). The sums of these series are entire functions(they are holomorphic on C”). We recall

We introduce the power series expansion

A ( z )=

c .,(-.)-?

n20

where

c, =

r(n + ;)r(n + i)(-),3 n!

(nE

4

N’);

cx, = 1.

Let t = z? with v E N’. We set, by definition, @[[t]ls = C[[z+]ls. If f^ E C[[z+]ls,we will say that it is of order vs in 2 in a generalized sense.

5

Theorem 1.6. (i) The power series A(,) E C [ [ z - i ] ]is Gevrey of order in $ and g-summable (in $) or 3-summable in t (= 2-3) in every direction except argt = -IT or &: (mod. 27~). (ii) The s u m of 4 z - 1 / 4 e - 2 / 3 r 3 ’ 2 ~ ( zin ) the direction R+ is equal to 4rr 2 the classical Airy function A i ( z ) . We set B ( z ) = C n > o ~ n = ~ A(-.). -% z-1/4e-2/3z3/2

AM,

z

Then --1/4e2/3z3/2

75

form a formal system of fundamental solutions of Airy ordinary differential equation at z = 00. By %-summation in a general (i.e. non singular) direction we get a system of actual fundamental solutions. The jumps when we move the direction correspond to Stokes phenomena (cf. 2.1 below)

2. Applications to ordinary differential equations

2.1. Linear ordinary differential equations, index theorem and Newton polygons We will study in this part spaces of solutions of linear analytic differential equations. We will give only the main results (for detailed proofs, cf. [25],

[261)* The situation is local and we are interested in solutions of an operator d D = a,(x)(z)m ao(x) ( w h e r e a o , . . . ,a, E C { x } ) in some (topological) vector spaces. An important result is the finiteness of the dimension of a solution space. Traditionally we can try to compute this dimension using a fixed point method. This is elementary but it can be technically difficult. Here we will use a different approach based on finite index operators (Fredholm Operators). It is less precise: it gives a lower estimate of the dimension (and not in general the dimension itself), but it is very easy to apply (due in particular to stability by compact perturbations).

+

+

Definition 2.1. Let E , F be two complex vector spaces. A linear map u : E 4 F has a finite index x(u)if dimc ker u and dimc cokeru = dimc F/imu are finite. Then x(u) = dim ker u - dim coker u.

Example 2.1.

D

x ( x ) = -1; d D =x(-) dx : @“x]]4 @ “ X I ] , dx = 1. Proposition 2.1. Let E , F, G three complex vector spaces and u : E + F, v : F 4 G two linear operators. W e suppose that u and v have a finite index. T h e n the operator v o u : E + F has also a finite index and x(v O u> = x ( v >+ x(u>. = x : @“XI] + @“x]],

d

Example 2.2. d . x(xqZ)z) =i d . -)% : @“x]]4 @ [ [ X I ] . x’(dx 3

76

Lemma 2.1. (F. Riesz) Let E be a complex Banach space. Let K E L ( E ,E ) be a compact linear endomorphism o f E . Then the operatoridE+K is a finite index endomorphism of E and X(idE K ) = 0.

+

Theorem 2.1. Let E , F be two complex Banach spaces. Let K E L ( E ,F ) be a compact operator. Let u : E -+ F be a continuous operator admitting a finite index. Then u+ K E L ( E ,F ) has a finite index and x ( u+ K ) = ~ ( u ) . Be careful: In general d i m ker(u

+ K ) # d i m ker u.

Lemma 2.2. Let El, F1 and E2, Fz be Banach spaces or Fre'chet spaces, or DFS spaces(l) (same type f o r each pair, but the two pairs are allowed to admit diflerent types )I3. If we have a commutative diagram

where u1, u2, v , w are linear continuous maps, v and w are injective and w is dense, then: (i) W e have an exact sequence

u1

are finite index maps,

-

0 -+ kerp2 -+ EZ/E1

3 F2/Fl

and ker uz has finite dimension with dim ker fiz = ~ ( 7 ~ -2 x(u1). ) This implies X ( U Z ) 2 x(u1). (ii) The pair ( v ,w)is a quasi-isomorphism (i.e. it induces isomorphisms o n the kernels and colcernels o f u l and u2 ) if and only if x(u1) = ~ ( u z ) . This lemma generalizes some results of Dieudonnk and Schwartz and was communicated to the author by B. Malgrange. We will now explain how to compute indices of a differential operator D using the Newton polygon N ( D ) of D. The Newton polygon in this form is due to the author [26]. (There are a lot of Newton polygons associated to differential equations are more general functional equations in the literature: Fine, Adams, Komatsu . . . ) Let D = a,(-&), . a. be a holomorphic (or meromorphic ) linear differential operator: ao, ' a , E C { x } (or C { x } [ x - ' ] ) . We write the

+ + a

e

.

13DFS spaces are dual of FS ( FrBchet Schwartz) spaces or equivalently DF spaces of Schwartz type.

77

expansion of a i ( x ) at the origin:

aij E C and aij = 0 if j 5 ni E Z. By definition the valuation w(ai) of ai is the smallest integer j such that aij # 0.

i-J

Fig 2.1.1 We can write D as an infinite sum of elementary operators a i j x j ( & ) i with aij # 0. If aij # 0, we put a dot at the point of coordinates ( i , j - i) (cf. Fig 2.1.1.). For each dot ( a , b), we translate the second quadrant at the point ( a , b ) and we draw the convex hull of all the quadrants (translated at all the allowed values of ( i , j - i) ): cf. Fig 2.1.2. We get the Newton polygon N ( D ) of D. The lowest point of N ( D ) corresponding to a fixed value of i is ( i ,w(ai)2 ) . The theory of index of O.D.E. begins with a result of Malgrange:

+

Theorem 2.2. (Malgrange) Let D = am(&)m t.. . a0 E C { x } [ & ] . Then: (i) D : C [ [ x ]-+] @[[XI] has a finite index:

(ii) D : C { x } + {x} has a finite index: Xan = m - w(a,)

78

Fig 2.1.2

(iii) W e consider D : @[[z]]/@{x}2 @[[x]]/C{x}. T h e n D is surjective, has a finite index, and dimkerD = x ( D )= XformalBy definition

Xformal

an.

- xan 2 0 is the irregularity index of D.

Corollary 2.1. The following conditions are equivalent: (i) D i s a Fuchsian operator (regular singular); (ii) N(D) i s a rectangle (i.e. it has n o strictly positive slopes); (iii) The irregularity index of D is zero. Remark 2.1. The formal index corresponds t o the lowest part (horizontal slope) of N ( D ) (with a change of sign). The analytic index corresponds t o the lowest point of the vertical slope of N(D) (with a change of sign) which is the dot marked for the symbol z+n) ( & ) m of D. The problem of irregularity of an operator was studied by many authors:

J. Moser, Levelt, Gkrard-Levelt, Malgrange, N. Katz . . . In [26], I introduced an interpolation between Malgrange analytic and formal indexes. The idea is t o use the interpolation between C{z} and C[[x]]by Gevrey spaces @[[x]ISof formal power series:

@{x} = @“.I10

c @“xIls c @“XI1

=

@“~ll~;

@[[z]IS increases with s E (0, +co).

Theorem 2.3. (Ramis) Let s E [O, +co]. W e set k (i) D : @[[x]lS-+ @[[x]IShas a finite index:

=

:. Then:

x s ( D )= i ( k ) - v ( a i ( k ) ) (cf. Fig 2.1.3)

79

Fig 2.1.3

(ii) dimker(D : @[[x]IS/@{x} -+ @ [ [ x ] ] s / C { zdecreases }) with s and is locally constant except for some ‘?jump points” corresponding t o the finite set of slopes of the Newton polygon N ( D ) .

If we fix k, then we can consider all the lines of slope k cutting the closed set N ( D ) (i.e. such that the intersection is not void). The smallest intersection is in general reduced to a point ( i ( k ) , v ( a i ( k )-) i ( k ) ) . We have an exceptional situation when k is one of the slopes of the Newton polygon N ( D ) ; in such a case the smallest intersection is a segment (the corresponding slope of N ( D ) ) and we choose for i ( k ) the smallest possible value. Proof of Theorem 2.3. We will give only the main ideas. When this result is due to 0. Perron who proved it by a delicate computation using a fixed point method. Our proof is the following: We write

D E

@[XI[&]

D

= a i ( k ) , ” ( % ( k , ) Z”(%.)) ( - L ) i ( k ) + D1 = Do + D1

+

in the “general case” and D = Do D1, where Do is the sum of the ai,jxj(&)Z where ( i , j ) belongs to the segment of slope k of N ( D ) in the exceptional case. Then we write @[[x]];as an union (inductive limit) of Banach spaces of type ll(C[[x]]g is a DFS space). The next step is to interpret D as a compact perturbation of DO (2.e. D1 as a compact operator) between some Banach spaces (much precisely we do that for a family of pairs of Banach spaces). It remains to compute x(D0)“by hand” (using a variant of Example 2.1.2 and if necessary Proposition 2.1.1 and a simple computation with Euler operator), to get x ( D ) = DO 01) = DO)

+

80

using theorem 2.1.1, and to conclude by an inductive limit argument based on Lemma 2.1.2. This proves (i). Using Lemma 2.1.2 and (i), we get a small exact sequence

0

+

Hs

+

5 @~[.:ls/{}

} . { @ / s I ] . “ @

+

0

where

Hs

= ker(D }:.{@/sI].“@

+

@“.lls/@{~})

The index x s ( D ) = dimc ker H, clearly decreases with s, remains locally constant except perhaps when s = corresponds t o a slope k of N ( D ) . It is semicontinuous on the left. We denote by kl > kz > . .. ke the strictly positive slopes (finite) of N ( D ) . We set s,. = ( r = 1, ..., t ) . Then H, = Hse if s 2 s>; if s E [s,.,s,.+l)then H, = H s r ; if s E [O,sl), then H , = (0). I f f E @[[z]]is a solution of Dy = g E @{z},then f^ E @[[.I], with s 5 sl; moreover, if s E [s,,s,+1) then we have exactly E @[[x]],,. Hence, we get the following result:

2

f”

Theorem 2.4. (Ramis) If D is a linear differential operator analytic in a is a formal power series solution neighborhood of the origin and i f f ” E @[ XI] of the inhomogeneous 0.D. E. Dy = g, where g E @{x} is convergent, then f^ is Gevrey of optimal Gevrey order s = 1+ i , where k is one of the strictly positive (finite) slopes of the Newton polygon N ( D ) of D .

f” @[XI],

A formal power series f^ is said of optimal Gevrey order s if E and f^ @ for any s’ < s. Maillet proved (1904) that a formal power of a (not necessarily linear) analytic O.D.E. series solution y = f^ E ~[[z]]

@[XI],

G(., Y,Y’,.. . 7 Y‘”9 is Gevrey of some order s, but his estimate for s is quite bad. Maillet theorem was rediscovered in the 50s by K. Mahler. Recently Malgrange extended theorem 2.4 t o the nonlinear case getting a new version of Maillet theorem with optimal Gevrey estimates; he uses a Newton polygon along the choosen formal solution.

2.2. Fundamental existence theorems Our aim is to “represent” a formal power series analytic O.D.E.:

G(x,y, y’, . . * ,y‘”’)

=0

f^ E @[solution [.I of an ]

81

by an actual solution y = f ( x )of the equation, holomorphic on a sector and asymptotic to f. As we explained before, the story begins with H. Poincar6. Many authors worked on this problem (Malmquist, Birkhoff, Hukuhara, * * . ), but the complete solution is quite recent [30]. Here we will mainly study the linear case in relation with Gevrey estimates, Ic-summability and Stokes phenomena. It is more easy to work with differential systems

(A)

Y’=A(x)Y

where Y is an unknown function of the complex variable x , taking its value in C”, and A a given ( n ,n) meromorphic matrix in a (small) neighborhood of the origin. It is easy to derive a system from an 0.D.E of order n in using the classical trick:

Y = (y, y’, . . . ,y@-l)). Conversely we can derive (non uniquely) an 0.D.E of order n: D, = 0 , from a system (A) using a “cyclic vector method” [26]. The Newton polygon N ( D ) is independent of the choices and we can set N ( A ) = N ( D ) . At the end of the century, Fabry got a fundamental system of formal solutions for an analytic linear 0.D.E at a singular point. For systems the result is due (independently) t o Hukuhara and Turrittin.

Theorem 2.5. (Hukuhara - Turrittin) Let

(A)

Y’=A(x)Y

be a (germ of) meromorphic differential system at the origin. admits a formal fundamental matrix solution

F

Then it

= fi(t)x’eQ(a)

where: t” = x (u E N * ) , L E End (rn;C)is a constant matrix, x L = e ( L O g z ) Lfi , E GL(m,C[[t]][t-’]) is a formal invertible matrix and Q = ( 4 1 , . .. , q m ) is a diagonal matrix where qi E $[+I (i = 1, ... , m ) (qi can be zero). Example 2.3. For the system of rank m = 2 associated t o Airy equation (Y = ( y ,y’); y” - ay = 0), we have u=2, t2 = x , 2 2 q1(t) = --t3 = - - x T 3 3

3

’

2 , q 2 ( t )= -t 3

=

2 3 -zT 3

82

In Theorem 2.5, the integer v is the LCM of the slopes of the Newton polygon N(A) (which belongs to Q+ ). The ramification x = ’t to (A) is a system (A,) in t ; the slopes of (A,) are the slopes of (A) multiplied by v ; they are integers (the degree in t of the polynomials qi ). It is very important to notice that in general the entries of the matrix H ( t ) are divergent series (like in the Airy example . . . ) In a sector V , we can represent the “symbol” x L by an actual holomorphic function: we write x L and we choose a determination of the logarithm over V . In the linear case we have the following fundamental existence theorem (Birkhoff fundamental existence theqrem) in PoincarB’s style.

Theorem 2.6. (Birkhoff) Let

(A)

Y’=A(x)Y

be a (germ of) meromorphic system. For each direction d , there exists a sector V bisected by d and a holomorphic matrix H in V , asymptotic in V to H ( i n Poincare’s sense) such that F = H x L e Q is a fundamental actual solution of ( A ) . We suppose that when we write the expression of F , we have chosen a branch of Logx on V and therefore a determination of x L and a determination of t = x t = e(t)Logz. The reader can find a proof of Theorem 2.6 in W. Wasow’s book [33]. The proof is quite delicate. It is possible to improve this result: if ke is the smallest strictly positive slope of N ( A ) ,then we can choose for V a sector of opening Moreover, for some choices of d it can be possible to choose a bigger sector [29]. We will give some indications below. Let q E $[+I : q ( t ) = pt-’++. ( p E C, k E N * ) . The “dominant part” of eq(t) is exp(pt-’). The directions (in the t-plane) such that pt-’ E iR are oscillation lines for exp(pt-’). They are (in the classical terminology) Stokes lines. Sometimes they are called anti-Stokes lines (which is more in the spirit of Stokes’ work. . . ). We suppose that the Newton polygon N ( A ) of our system admits only one strictly positive slope k > 0. Then qi = pzt-‘ ( i = 1,.. . ,rn). The pi’s are solutions of an algebraic equation (the indicia1 equation associated to the slope k). The simplest case is when the rn values of pi are all distinct. In this case, we can consider the system

2.

(End A)

X’

=AX

-

X A = [A,X ]

83

where X is an unknown (m,m) square matrix. The exponentials appearing in its formal fundamental solution are the qij = qi - q j = ( p l - p j ) t - k

( i ,j = 1 , . . . ,m ) ,

therefore k is the only strictly positive slope of N(EndA). For each pi E { P I , . . . ,p m } different from 0, we get 2k Stokes lines. (For Airy case, we get k = 3, m = 2, therefore 12 Stokes lines in the t = x i -plane. For Euler case, we get k = 1, therefore 2 Stokes lines.) The set of non zero values of pi has m’ < m elements. We have m’ families of Stokes lines (2m’k Stokes lines).

Theorem 2.7. (Birkhoff, Horn) Let

(A)

Y’=A(x)Y

be a ( g e m of) meromorphic system. W e suppose that N(A) admits only one strictly positive slope k > 0 and that the m values pi are all distinct. To each n o n zero value of pij = pi - pj corresponds a family of 2k Stokes lines of N(End A). Then, i f a sector V contains at most one Stokes lines f o r each family ( p i j # 0 ; i , j = l , . . . , n ) , there exists a (m,m)square holomorphic matrix H o n V , H asymptotic t o H ( i n PoincarL’s sense) such that F = H x L e Q is a fundamental solution of A in V . Idea of Proof. We start from a smaller subsector W c V such that we can apply Birkhoff fundamental existence theorem (Theorem 2.6). We get a fundamental solution F1 = H l x L e Q , H1 H on W . Then we can extend analytically H I and therefore Fl on a bigger sector (We are dealing with linear equations). When we cross the first Stokes line which does not belongs to W , it can happen that the existence of H1 is no longer asymptotic to A. If this is the case we modify F I , multiplying it by a well chosen constant matrix C E End (m;C) on the right. We end the proof iterating this process. When we have crossed a Stokes line of each family we have loosed all the freeness for a convenient choice of C and it is impossible to cross a new line in the extension process. This corresponds to a result of uniqueness.

-

Theorem 2.8. In the situation of Theorem 2.7, zf the sector V contains exactly one Stokes line f o r each family, then the fundamental solution F is the unique solution asymptotic t o f (i.e, H H ) o n the sector V .

-

84

In the setting of O,D.E., Birkhoff fundamental existence theorem follows from Bore1 - Ritt theorem and the following result (cf. [33]).

Theorem 2.9. Let D E C { x } [ & ] be a linear differential operator. Then 0), we can we have d 1 and c is a positive constant. There exists a unique solution to (1.7) when N = 1 and N = 2, while for N 2 3 we require that p < p, (cf. [65]. The one-dimensional problem with N = 1, plays an important role in our survey. In this case, with p = y, we readily calculate that

For the special case p = 2 and N = 1, we get w(y)= $ech2(y/2). The topics in this survey are organized into sections of increasing problem complexity. In Sec. 2 we consider scalar nonlinear problems. We first focus on constructing k-spike equilibria to nonlinear boundary value problems of the type originally considered in 1131. Then, we describe some results for spikes in scalar quasilinear elliptic problems. In Sec. 3 we study the stability and dynamics of spikes for certain scalar, but nonlocal, problems. One such problem arises from the shadow limit associated with (1.1). In Sec. 4 we survey some bifurcation and dynamical phenomena that occur for one-spike solutions to the reaction-diffusion system (1.1). Finally, in Sec. 5 we give some equilibrium and stability results for multi-spike patterns of the reaction-diffusion systems (l.l),(1.5), (1.3), and (1.6).

2. Spike Equilibria for Scalar Problems In this section we begin by constructing asymptotic solutions for E~

u"

+ Q ( u )= 0 ,

-1

< z < 1,

103

with various boundary conditions at 2 = f l . The assumptions on Q ( u )are that Q ( u ) is smooth and that it has only two zeroes, s and s b , with s < S b , where Q ' ( s ) < 0, and Q ' ( s b ) > 0. In addition we assume that there exists a u, > s b such that Q(q)d q = 0. In Fig. 1 we plot such a Q ( u ) ,and in Fig. 2 we plot the corresponding phase-plane u, versus u,where z = Z / E , showing a saddle point at u = s and a separatrix structure.

ssu"

Figure 1. Plot of Q ( u ) versus u with Q ( s ) = 0, Q'(s) < 0 , Q(Sb) = 0, and Q'(sb) > 0. There, exists a urn > S b such that Q ( q )dg = 0. Here s = -1 and sb = 1.

s,""

The separatrix corresponds to a homoclinic solution w(y) to (2.1) on -00 < y < 00, with W(*CO) = s. Up to an arbitrary phase-shift, w(y) satisfies W"

+ Q ( w )= 0 ,

w'(0)= 0 , w(0) > 0 ; w(y) Here, B

< y < 00 , s + ce-OY , lyl + rn .

-CO

(2.2a)

(2.2b)

> 0 and c > 0 are defined by

sJm

where V ( q )= Q(0)do. This homoclinic solution leads to the existence of spike layer solutions for (2.1), where the spikes are localized near certain points in the domain. Depending on the boundary conditions for (2.1), there may be boundary layers near one or both endpoints that correspond to pieces of the homoclinic orbit. A phase-plane analysis of (2.1) for various boundary conditions was made in [84] and [88]. The main observation is that by using the stretching

104

-251 -25

"

-2

-15

'

-1

"

4 5

0

"

05

1

' l5

'

2

25

Figure 2. Plot of the phase-plane uz versus u,with z = X / E , for a Q(u) that admits spike-type solutions.

y = X / E in (2.1), it is clear that boundary and internal layer solutions for (2.1) correspond to trajectories in the phase-plane that are close to the separatrix in Fig. 2 and are away from the saddle point region near u = s. Since the solution spends a long time near the saddle point and a comparably shorter time to make a complete or partial circuit around the separatrix, it is clear that u s on -1 5 x 5 1 except for O ( E )transitions, or spike layers, located near the endpoints or at some interior points in the domain. This geometrical picture is very useful for obtaining qualitative information regarding the admissible spike configurations. In particular, it shows that for the boundary data u = s at x = f l , solutions with two or more interior spikes are impossible for (2.1) since any trajectory must necessarily lie outside the closed region bounded by the separatrix. Despite the simplicity of the geometrical picture afforded by the phase-plane, it does not readily yield detailed quantitative information regarding questions such as determining the locations of the spikes for arbitrary boundary data, the possibility of bifurcation behavior, and determining the number of solutions that exist for a fixed but small E . Furthermore, the phase-plane does not generalize to the singularly perturbed quasilinear elliptic problems in two or more spatial dimensions. Therefore, it is desirable to develop an asymptotic method to construct spike layer solutions to (2.1) that can be extended to the multi-dimensional case. The first example illustrating the difficulty in applying the method of matched asymptotic expansions to treat (2.1) was given by Carrier and Pearson (see pages 202-205 of [13]) for the special nonlinearity Q(u) = N

105

u2 - 1, and with u ( f 1 ) = 0. For this reason, we refer to (2.1) as Carrier’s problem. Near each endpoint there are two possible boundary layer solutions. Superimposed on these boundary layer solutions, they tried to construct a solution with one interior spike. However, as shown in [13],a routine application of the method of matched asymptotic expansions fails to determine the interior spike location 50. In the vicinity of z = ZO, it is easy to see that u has the form

u ( . )

N

2w

[&-1(2

-

where w(y) is given in (1.8) with p = 2. This failure in determining 50 is not restricted to the choice Q(u)= u2- 1but is typical for the class of problems (2.1). An extension of the method of matched asymptotic expansions was used in [66]to determine the spike locations for Q(u)= u 2 - 1. There it was shown that the failure of a routine application of the method of matched asymptotic expansions in determining the spike locations was a result of ignoring exponentially small terms in the expansion of the solution. By extending this method to retain and match the dominant exponentially small terms, it was shown how to find the correct spike layer locations for Q(u)= u2-1. Another analytical approach to determine the spike locations for Q(u)= u2 - 1 was given in [51]. They employed a variational principle, with trial functions from the matched asymptotic expansion solution, and determined the spike locations by making the variation stationary with respect to the spike layer locations. More recently, for the nonlinearity Q ( u ) = u2 - 1 a rigorous shooting method has been developed in [85] for constructing spike layer solutions, and for determining the number of such solutions for a fixed E with E 0 and is O( 1) as E 4 0. This problem models the initial formation of boundary layers near the endpoints. For the special case Q ( u ) = u2 - 1, and for two interior spikes, such a problem was analyzed using variational methods in [51], where it was shown that a bifurcation can occur if A exceeds a threshold. The general case for arbitrary n 2 2, and for arbitrary Q ( u ) of the form shown in Fig. 1, was treated using the projection method in $6.1 of [loo], where the following result was obtained:

Proposition 2.6. Let E 2c, two sets when A,(n) < A < 2c, and no such sets when A < A,(n). Here A,(n) is defined b j n-1

As(~)=~C(%)

(1-n) / 2 n

(x) ,

n=2,3,.

(2.41)

Therefore, by perturbing the boundary conditions by exponentially small terms, new solutions can be created by saddle-node bifurcations. To illustrate this result, let Q ( u ) = u2 - 1. Then, c = 12, and A,(2) = 21.06 from (2.41). Therefore, for E 0, ri = K(X;E ) has the form

+ 2'g(x)e-E

-1

(2.42b)

4.

Here g ( x ) is a smooth function. In [46] the projection method was used to determine the location of a spike for an interior one-spike solution with no boundary layers. The result is summarized as follows:

Proposition 2.7. Let E 1. We assume that Q ( s ) = 0, with Q’(s) < 0, and that there exists a unique radially symmetric ground-state solution w ( p ) ,with p = Iyl, that satisfies

w

//

+-( N P. - 1 ) w + Q ( w ) = 0 ,

w’(0) = 0 , w

I

N

s

+ cp-(N-1)/2

e- u p

p L0;

,

w(0) > 0 ,

(2.47a) (2.4713)

as p + m ,

. An important example is Q ( u ) = --u+u,p > 0 and = -Q’(s) Ill2 is the Sobolev exponent for N 2 3 , and for 1 < p < p , , where p , =

[

where c

p , = m if N = 2. For this case, w ( p ) satisfies (1.7). Equation (2.46) is the multi-dimensional version of Carrier’s problem (2.1), where w ( p ) replaces the homoclinic solution w ( y ) . The study of spike solutions to (2.46) was largely initiated in the pioneering work of Ni and Takagi (cf. [68], [76], [77]). An earlier survey of results for (2.46), and for some related problems, is given in [78]. We now follow [78] and give a rough summary of the results of [76] and [77],characterizing the “least-energy solution” of (2.46) for Q ( u ) = -u+uP, and with u > 0 in R. For this problem, the energy functional for (2.46) is

where u+ = max(u, 0). As argued in [76] and [77], J, has a minimum when restricted to the set of solutions of (2.46) with u > 0 in R. This minimizing solution is called the “least-energy solution”. Since an interior spike solution has, asymptotically, twice the energy of a boundary spike solution, the least-energy solution must be a one-spike solution centered at some point on d o . To determine the actual point (, E dR where the spike concentrates, a two-term expansion for JE as E 4 0 is required. For a spike centered at E 80, it was shown in [77] that

c,

(2.49) where C is a positive constant independent of

I(w)=

LN[;

E,

and I ( w ) is defined by

(IVWl2+ w 2 ) - -WP1

P+l

1

dy.

(2.50)

120

Here H (&) is the mean curvature of dR at following result was proved in [76], 1771:

&. By minimizing (2.50), the

Proposition 2.8. Consider (2.46) with u > 0 in R and Q(u)= -u + u p , where 1 < p < p,. Then, for E 0, there exists boundary Ic-peak solutions to (2.46) where the peaks all cluster near a local minimum point of H ( P ) . This clustering effect is qualitatively similar to the spike-clustering phenomena for the onedimensional problem (2.45) described in Sec. 2.4. Boundary spike solutions, together with asymptotic estimates for the eigenvalues of the linearization that tend to zero as E + 0, are also given in [lll]and [4]. Next, we describe some results for interior k-spike solutions to (2.46). These solutions have the form k

(2.51) j=l

for some E R for j = 1 , . . . , Ic to be found. As for Carrier’s problem in Sec. 2.1-2.3, since w(Iyl) decays exponentially as IyI -+ m, the linerarization of (2.46) around an interior spike solution is exponentially ill-conditioned. For an interior one-spike solution, it was shown formally in [loll, and proved rigorously in [110], that the spike concentrates at a local maximum of the distance function. The result and method used in [loll is described more precisely below. Geometrically, this one-spike result for (2.46) is asymptotically equivalent to that given in Proposition 2.2 for the one-dimensional problem in the sense that an interior spike for (2.46) concentrates at a point in R that is farthest from the boundary. In [41] a solution to (2.46) that has one boundary spike and one interior spike was constructed. It was found that the location of the interior spike is moved an O(1) distance from the center of the largest inscribed sphere for R in the

121

direction away from the boundary spike. Such a mixed boundary/interior spike solution is the multi-dimensional equivalent of the result in Proposition 2.5 for Carrier’s problem, where for E’U” + Q(u)= 0 with u(-1) = U L and u’(1) = 0, an interior spike is located not at the midpoint 20 = 0 but instead at 20 = 1/3. For an interior k-spike solution of (2.46), with k 2 1,the following result with relatively minor technical differences, was given in [59], [40], and [3]:

Proposition 2.9. Equation (2.46) admits an interior k-spike solution given asymptotically by (2.51), where the concentration points e l E , .. . , tend to local maximum points of $(El, . . . , &) as E -+ 0, where

$(ti,.. . ,&) = i , j , l =min l ,...,k l j # l Notice that this result is geometrically very similar to the analogous result in (2.24) for a k-spike solution to Carrier’s problem with Neumann boundary conditions. The main difference in the multi-dimensional case is that, depending on the topology of 0, there can be many different choices for the set of spike locations. F’rom Proposition 2.9, it is clear that the spike locations are asymptotically equivalent to a corresponding geometric ball-packing problem. The next result, given in Corollary 1.8 of [3], makes this equivalence precise.

Proposition 2.10. Let 5’1,. . . ,s k E 0 be nonoverlapping spheres of the same radius d, and assume that 5’1,.. . ,s k are packed in such a way that when considered as rigid bodies in a rigid container 0 , the set &, . . . ,& of their centers becomes also a rigid body. Then, for E > 0 sufficiently small, (2.46) has a solution with k spikes that localize at (1, . . . , &. In Fig. 2-6 of [3], many illustrations of this “rigid-body” geometrical construction are shown. In particular, in Fig. 5 of [3], several possibilities are shown for packing eight small spheres of a common radius inside a spherical domain R. Although the basic theory for spike solutions of (2.46) is rather well established, there are two questions that should be explored.

Question 2.6. Formulate a numerical method to compute interior k-spike solutions to (2.46) for E l?

122

Question 2.7. Using techniques in computational geometry, investigate how the topology of R influences the geometrical ball-packing characterization given in Proposition 2.10 for a k-spike solution of (2.46) with k large. Can one estimate the number of solutions for E small but fixed? In [loll, a multi-dimensional extension of the projection method, as outlined in Sec. 2, was used t o determine an interior spike location. We now sketch this method for a one-spike solution of

E ~ A+UQ ( u ) = 0 ,

5

E

R;

E~,U

+ b(u -

S)

=0,

x E do.

(2.53)

Here R E B2, and b = b(6) 2 0, where 6 is arclength along the smooth boundary dR. We look for an interior one-spike solution of the form u N w [ ~ - ' l x- xol] R, where R -1 f o r i = 1’2, where bi = b(&) and where tci is the curvature of dR at ti. Then, xo lies o n the chord joining ~ ( 6 1 and ) x ( & ) . Moreover, f o r E + 0 , xo satisfies Xo(E)

= xin

+ -x;fi2 +O(E2)’ 8a E

(2.61a)

where

Notice that if (bl - a)(b2 - a ) < 0, then (2.60) has no root near xin. This condition is qualitatively similar to the condition given in Proposition 2.2 for Carrier’s problem. A similar result for when the largest inscribed circle makes three-point contact with dR was given in [loll. There are several rigorous results for an interior spike solution for (2.53). For the Dirichlet problem with b = 00, it was proved in [79] that there

124

is a least-energy solution where a one-spike solution concentrates at the maximum of the distance function. This result can be obtained by letting b -+m in (2.61). In [S] an interior spike for 2.53) was analyzed in a halfspace when b is near the critical value b = rs. As b -+ CT+, the spike was found to approach the boundary. The sensitivity of the spike location for b near rs is certainly suggested from (2.61b). For the Neumann problem, the result in Proposition 2.10 shows that there is a plethora of interior spike solutions. However, as for Carrier's problem in one spatial dimension, interior multi-spike solutions for the Dirichlet problem with u = s on dR should not exist. This leads to the next question.

Question 2.8. What are the bifurcation properties of interior Ic-spike solutions for (2.53) with Ic 2 spikes under the Dirichlet boundary conditions u = s Ae-"€-l for some a > 0 and A > 0. When A = 0, there should be no such solutions. Do the solutions have a saddle-node bifurcation behavior similar to that described in Proposition 2.6 for Carrier's problem?

+

>

Question 2.9. For general Dirichlet data with u = u b ( J ) > s on dR can one construct a solution to ~ ~ n u + Q (= u 0) that concentrates on the entire boundary of dR and that has Ic 2 1 interior spikes? We conjecture that, in analogy with Proposition 2.4, the interior spikes now concentrate at local maximum points of (2.62)

A related problem where localization occurs is for the nonlinear Schrodinger equation (cf. [38], [22] and [99]) €2Au-V(Z)u+uP=O,

zER;

a,u=o

ZEdR.

(2.63)

Here V(z) is a smooth positive potential with V(z) > V , > 0 in 0, and p is subcritical. This is the multi-dimensional counterpart of the modified Carrier problem (2.45). It is well-known that there exists spike solutions of (2.63) that localize near non-degenerate local maxima and minima of V(z). Equation (2.63) also admits spike-clustering phenomena where Ic spikes all cluster near a minimum of V(z) (cf. [99]). It would be interesting to compare the spike phenomena for (2.63) for an exponentially shallow potential with that for the quasilinear problem (2.46). This leads to the next question.

Question 2.10. What are bifurcation properties of spike solutions for (2.63) for a potential of the form V(z) = 1 e-"/'V(z), where rs > O?

+

125

The spike locations should be determined from a competition between the distance function and the localizing effect of the potential p ( x ) . Another problem where localization occurs is in the construction of hotspot solutions for Bratu's problem

Au+Ae"=O,

X E R ; u=O, X E ~ R .

(2.64)

Here 0 is a bounded, simply-connected, domain in R2. The qualitative feature of hot-spot solutions is that u 4 00 as X 4 0 in a localized region near some x = i$,for j = 1 , . . . , k while u = 0(1)as X 4 0 away from these points. Using complex analysis, a system of equations for the hotspot locations &, for j = 1,.., k, was derived in [75]. An alternative method based on singular perturbation theory was used in [loll. The following result characterizes the hot-spot locations: Proposition 2.12. For E satisfy the coupled system

+ 0,

the hot-spot locations

61,. . . , & for

(2.64)

Here G d ( x ; ( ) is the Dirichlet Green's function, with Rd(x; 0. Since the coefficients in the differential operator depend only on y = E - ~Ix - xoI , we look for localized eigenfunctions @(y),which decay as IyI -+ 00. Therefore, it is natural to try to compare the spectrum of (3.8) with that of

S"

M o @ ~ A @ + Q , @ + Q , E ~ g,@dy=X@,

Y E V @.-to ,

y--+00.

(3.9) Here the derivatives are with respect to the y variable. This problem is referred to as the infinite-line nonlocal eigenvalue problem. We first note that the spectrum of (3.9) has N zero eigenvalues with corresponding eigenfunctions @j(y) = ay,uq(JyI), for j = 1 , . . . , N . For these functions the nonlocal term in (3.9) vanishes identically since gu is radially symmetric in IyI. As a result of the exponential decay of u p ,the discrete

129

eigenvalues of (3.9) should be exponentially close to corresponding eigenvalues of the finite-domain nonlocal eigenvalue problem (3.8). This suggests that there are N eigenvalues of (3.8) that will be exponentially small, and whose eigenfunctions $ j E can be approximated by $ j E = uq $ b j , where $b is a boundary layer function localized near dR that allows the no-flux condition (3.8b) to be satisfied. Notice that the boundary layer calculation is in the same spirit as that done in Sec. 2.1 for Carrier’s problem. Secondly, we note that if we neglect the nonlocal term in (3.9), the resulting local eigenvalue problem will have one eigenvalue that is strictly positive corresponding to an eigenfunction @p1 that is of one sign. Since in gu@pl d y # 0, the nonlocal term in (3.9) will perturb this eigengeneral pair significantly. The key step in the analysis is reduced to determining whether the nonlocal term in (3.9) is sufficiently strong to push this positive eigenvalue associated with the local problem into the left half-plane Re(A) < 0. Since (3.8) only perturbs this eigenvalue by exponentially small terms, it remains strictly in the left half-plane for the finite-domain nonlocal problem. If this spectral condition holds, it would follow that an interior one-spike equilibrium solution is metastable in the sense that the eigenvalues in the spectrum of the finite-domain nonlocal problem (3.8) that have the largest real parts are exponentially small as E -+ 0. The corresponding eigenfunctiom are closely approximated by the translation modes d Y J u q ( l y l )for , j = 1,.. . , N . This rough sketch outlines the mechanism through which the nonlocal term can eliminate one unstable eigenvalue of the corresponding local eigenvalue problem and ensure stability on an 0(1)time-scale. Depending on the sign of the exponentially small eigenvalues, an interior one-spike solution may not stable on an exponentially long time-scale. However, these exponentially small eigenvalues will lead to the existence of a metastable time-dependent behavior for an interior one-spike solution. As mentioned above, the key step in the analysis is to find conditions for which there are no eigenvalues of (3.9) with Re(A) > 0. In general, eigenvalue problems of the type (3.9) and (3.8) are difficult to analyze since they are in general not self-adjoint, and hence complex eigenvalues are possible. To illustrate this possibility, consider the eigenvalue problem (3.8) in one space dimension when R = [-1,1] and 50 = 0. The resulting problem has the general form

ax, +

130

with &(fl) = 0. Here S is a parameter measuring the strength of the nonlocal term. This eigenvalue problem is not self-adjoint unless B ( z ) = k C ( z ) for some constant k. For fixed E , many properties of self-adjoint eigenvalue problems of the class (3.10) were obtained in [35] and [9]. Consider the example of [50] where E = 1, A(z) = 0, and

1 1 ~ . ~ - - C O S ( T Z ) + - C O S ( ~. T Z ) 2 2 (3.11) Moveable eigenvalues are those eigenvalues of the local problem that are perturbed by the nonlocal term. Fixed eigenvalues refer to those eigenvalues of the local problem that remain independent of 6, since their eigenfunctions are orthogonal to C(z). For this example, the only moveable eigenvalues are those for which the eigenfunctions lie in the subspace spanned by

B ( z ) 2.5+~0s(nz)+2CoS(27r2) C(Z)

4 = so + s1 cos (7rz) + s 2 cos (27rz) ,

(3.12)

for some s o , s1, and s2. Substituting (3.11) and (3.12) into (3.10) where = 1, we get the matrix eigenvalue problem (A - bD) s = As, where

E

0 0

0 0 0 0 -4n2

A = (0

-7r2

)

, D=

5.5 -1.25 1.25 (2.2 -0.5 0.5) 2.2 -0.5 0:5

,

s

=

(sn)

. (3.13)

The real parts of the eigenvalues as a function of b are shown in Fig. 4. In this figure, the dotted lines correspond to the fixed eigenvalues -k27r2/4 for k = 1 and Ic = 3, corresponding to the eigenfunctions q5 = cos (k7r(z 1)/2) for k = 1,3. This simple example shows that nonlocal non self-adjoint eigenvalue problems of the form (3.10) can have complex eigenvalues through the collision of two moveable eigenvalues. An important class of nonlocal infinite-line eigenvalue problems that arises in determining the stability of spike solutions in several different systems is the following nonlocal non self-adjoint problem:

+

(3.14a) Here w(Jy1) satisfies (1.7), and Lo is the local operator

Lo@E A@ - @ + p w p - l a .

(3.14b)

We assume that m > 1 and 1 < p < p,, where p, is the critical Sobolev exponent. Notice that d,,w()y)) lies in the kernel of MO for j = 1 , . . . ,N ,

131

........................................................

...........................................................

Re($0

-30 -40

t

1

1 0.0

1.0

2.0

3.0

4.0

5.0

6 Figure 4. The real parts of the eigenvalues of (3.13) (solid curves) versus 6. Two of them are complex when 1.076 < 6 < 3.970. The dotted lines are two fixed eigenvalues X = -7r2/4 and X = - 9 7 ~ ~ 1 4not , contained in (3.13), which are independent of 6.

and so X = 0 is a fixed eigenvalue. There are two key formulae for Lol obtained by a direct calculation, that are needed below (3.15) For the one-dimensional case N = 1,where w(y) is given explicitly in (1.8), the following spectral results for (3.14) hold: Proposition 3.3. Let @ E H 1 ( R ) , and consider any nonzero eigenvalue A0 of (3.14). Then, we have the following:

For 0 5 a < p - 1 we have Re(X0) > 0 . Now suppose a > p - 1. Then, if either m = 2 and 1 < p 5 5, or, m = p + 1 a n d p > 1, we have Re(X0) < 0 . If p > 1 and m = p , then we have Re(X0) < 0 when p - 1 < Q 5 p . The proof of the first result for 0 5 a < p - 1 is given in Appendix E of [47]. The proof of the second result for a > p - 1 is given in Lemma A and Theorem 1.4 of [log]. The third result is proved in Theorem 1 of [115]. For the multidimensional case where N > 1, the following results are known: Proposition 3.4. Let @ E H1(RN), and consider any nonzero eigenvalue XO of (3.14). Then, we have the following:

For 0 5

Q

< p - 1 we have Re(X0) > 0 .

132

Now suppose a > p-1. Then, i f either m = 2 and 1 < p 5 1+4/N, or, m = p 1 and 1 < p < p,, where p , is the critical Sobolev exponent, we conclude that Re(Xo) < 0. Letm=p. I f 2 9 1 5 whenN=2, o r 2 1 p 1 3 whenN=3, then we have Re(X0) < 0 when a = 2p.

+

The proof of the first two results are given in [108], and the proof of the third result for a = p is given in Theorem 5.6 of [91]. Notice that when m = p 1, the operator is self-adjoint. Qualitatively, these results show that the nonlocal term may eliminate the unstable eigenvalue of the local operator only when a is large enough. We now comment on the bounds in these results. The lower bound a = p - 1 for stability in the second result of Propositions 3.3 and 3.4 cannot be improved since from (3.15) we readily calculate that Mow = 0 when a = p - 1. The upper bound for a in the third result of Proposition 3.3 is not sharp as stated in [115]. The upper bound on p for m = 2 in the second result of Proposition 3.3 is indeed sharp as the next result shows.

+

Proposition 3.5. Let E H 1 ( R ) , m = 2, and suppose that p > 5 in (3.14). Then, there exists a n a , with a , > p - 1 such that there are exactly two positive real eigenvalues in the interval (0,vo) f o r any a with ( p - 1) < a < a,. I n addition, there exists a value ah, with ah > am such that f o r am < a < ah, there as a pair of complex conjugate eigenvahes in the unstable right half-plane Re(X) > 0 . W h e n a = ah, there is a pair of complex conjugate eigenvalues on the imaginary axis. This result was proved in Proposition 2.7 of [50]. In addition, a detailed numerical study of the spectrum of (3.14) for different values of m and p was given in Sec. 2.2 and Sec. 2.3 of [50]. For any p 2 3 and with a = 0, Proposition 3.2 shows that there is only one discrete nonzero eigenvalue of Mo. Proposition 3.5 shows that there are two discrete eigenvalues in the right half-plane for some range of Q when p > 5. The numerical computations of [50] show that an extra eigenvalue is created out of the edge of the continuous spectrum at a certain value of a. The two discrete moveable eigenvalues then coalesce producing a complex conjugate pair as in the simple example (3.13). This leads to the next question.

Question 3.2. Find other ranges of p , m, and N where any nonzero eigenvalue of (3.14) will have Re(X) < O? Can one characterize any edge bifurca-

133

tions for (3.14) from the continuous spectrum? A detailed numerical study for N > 1 is an open problem. The analysis leading to Proposition 3.3-3.5 relies rather heavily on special properties of the nonlinearity Q(u)= -u+uP, most notably the explicit formulae (3.15). This leads to the following question.

Question 3.3. Can one characterize the discrete spectrum of more general problems of the form (3.9) around an interior one-spike solution? Although the proofs of Propositions 3.3-3.5 are too involved to discuss here in detail, we can still give a qualitative idea on how some of these results are obtained. To do so, we reformulate (3.14) by letting $(y) be the solution to

Lo$

= $" - $ +pwP-'$

= A$

+ wp ;

0 a~ IyI

$ 3

4 00.

(3.16)

Then, the eigenfunctions of (3.14) can be written as

(3.17) We then multiply both sides of (3.17) by wm-' and integrate over RN. Assuming, that wm-'@ dy # 0, we then obtain that the eigenvalues of (3.14) with even eigenfunctions are the roots of g(X) = 0, where

SRN

The function g(X) is analytic in the right half-plane except at the simple pole X = vo, where vo is the unique positive eigenvalue of Lo. A simple calculation of the winding number shows that the number A4 of zeroes of g(X) in Re(X) > 0 is

(3.19) Here [argglr1 denotes the change in the argument of g(X) along the semiinfinite imaginary axis l?I = i X I , 0 XI < 00, traversed in the downwards direction. Therefore, to calculate M , we need only determine properties of g(X) on the positive imaginary axis. We let X = XI and we separate real and imaginary parts by writing g ( i X I ) = ~ R ( X I ) i a j ( X ~ )A . simple calculation shows that the eigenvalues


0 f o r XI > 0.

+

+

The local behavior of f~ was derived in Eq. (4.3) of [106].The condition < 0 for m = 2 was derived in the proof of Theorem 2.3 of [106]. The local behavior off1 was derived in Eq. (4.2) of [106].The proof that f~> 0 for m = 2 and 1 < p 5 1 4/N is rather difficult, and was obtained in Theorem 2.3 of [106]. The condition fI > 0 for m = p 1 and 1 < p < pc is readily seen by writing f~as f~((x,) = XIC(XI), where

f;

+

+

(3.23)

135

We readily calculate using (3.15) that (3.24a)

C(O)=P -l1 " P - 1

"1

(3.24b)

2(p+1)

Thus, for 1 < p < p,, we have that C(0) > 0 together with C'(X1) < 0 for XI > 0, and C(X1) + O+ as XI -+ 00. Hence C(X1) > 0 for XI > 0, which establishes that f; > 0 for XI > 0 when m = p 1 and 1 < p < p,. Next, we use the properties of g on the imaginary axis to calculate M from (3.19). The following result is readily derived by using (3.20)-(3.22) to calculate [arggIr, :

+

Proposition 3.7. Let a > p - 1. Suppose that at each root of 6~ = 0, we have that f~> 0. Then, M = 0 , and there are n o eigenualues of (3.14) in Re(X) > 0. Alternatively, suppose that 0 < a < p - 1, and that f~ is monotone decreasing f o r XI > 0 . Then, M = 1 and so there is a unique real positive eigenualue of (3.14). Notice that if a > p - 1, then i j ~ ( 0< ) 0 and Gl(0) = 0. As A 1 + 00, we have i j ~-+ a-' > 0 and 61 4 0. Hence, if whenever we have a root of i j ~ = 0 it follows that 51 < 0, we conclude that [argglr1 = -r, and consequently M = 0 from (3.19). Note that 61 < 0 is guaranteed whenever f; > 0 for all XI > 0. As seen in Proposition 3.6, this condition is guaranteed for two cases: m = 2 and 1 < p 5 1 4/N, or, m = p 1 and 1 < p < p,. This criterion then establishes the second statements in Proposition 3.3 and Proposition 3.4. Alternatively, if 0 < a < p - 1 and < 0, then i j >~ 0 for X I > 0. Consequently, [arggIrr = 0, and hence M = 1. This is the first statement in Proposition 3.3 and Proposition 3.4 under a slightly weaker hypothesis. Eliminating the hypothesis that is monotone decreasing, it is readily seen, upon looking for roots of g(X) = 0 on the positive real axis, that M 2 1 when 0 < a < p - 1. Finally, we comment on the idea behind Proposition 3.5. For m = 2, we have from Proposition 3.6 that < 0 for XI > 0, and hence there exists a unique root to i j =~ 0 when a > p - 1. If we can guarantee that tjr > 0, or equivalently f~< 0, at this root, then we have [arggIrI = +7r, and so M = 2. For N = 1, m = 2, and p > 5, the local behavior in Proposition 3.6 shows that .f~< 0 for XI > 0 sufficiently small. Hence, there is some range of a with a > p - 1 for which M = 2. This is the essence of Proposition 3.5.

+

+

fk

FR

fk

136

The result in Proposition 3.7 gives a simple criterion to determine sufficient conditions for nonzero eigenvalues of (3.14) to satisfy Re(X) < 0. This leads to the following question.

Question 3.4. Can one find other ranges of m, p , and N, to ensure that the positivity condition on f~ given in Proposition 3.7 holds? With this condition any nonzero eigenvalue of (3.14) has Re(X) < 0 when (Y > p - 1. One of the earliest analyses of metastability for a shadow system was given in [64]. Another activator-inhibitor system that exhibits metastability was given in [5] and [6]. In the next few subsections we give a few explicit examples of the stability and dynamics of spikes for shadow systems.

3.1. The Shadow Gierer-Meinhardt Model Our first example of a shadow system is obtained by letting D -+ m in the GM model (1.1) to get at = E2Aa - a

+ ap/hq, Tht

x E R; =

-h+

&a

E - ~ I R(

=0

,

am Fdx.

x E dR ,

(3.25a) (3.25b)

In (3.25b), Is11 denotes the volume of 0. In this section we will consider the case where the reaction-time constant T in (3.25b) is zero. The possibility of Hopf bifurcations when T > 0 is discussed in the next section. An interior one-spike equilibrium solution to (3.25) in RN is given by

Here C is given in (1.2), W N is the surface area of the unit N-dimensional sphere, and w ( p ) satisfies (1.7). The finite-domain nonlocal eigenvalue problem of the form (3.8), obtained by linearizing (3.25) around this equilibrium solution, is

where an& = 0 on dR. The corresponding infinite-line nonlocal eigenvalue problem is

137

with + 0 as IyI 4 00, and L O defined in (3.14b). From (3.14), and the condition (1.2) on the exponents ( p ,q , m, s ) , we see that Q = m q / ( s 1) > p - 1. From the second statement of Proposition 3.4 we conclude that for any nonzero eigenvalue of (3.28), we have that Re(X) < 0 when m = 2 and 1 < p 5 1 4 / N , or when m = p 1 and 1 < p < p,, where p , is the critical Sobolev exponent. Therefore, under these conditions, the nonlocal term has pushed the unstable eigenvalue of the local operator LO into the stable left half-plane. Since the discrete eigenvalues of (3.28) are exponentially close to corresponding eigenvalues of (3.27), we conclude from the discussion following (3.8) that an interior one-spike solution to the shadow GM model will be metastable. Then, by using the projection method in a similar way as was done in Sec. 2, the derivation in Sec. 2.5 of [44] yields the estimate (2.58) for the exponentially small eigenvalues of (3.27). Therefore, to leading order, the contribution of the nonlocal term in (3.27) is subdominant to that of the boundary layer calculation given in Sec. 2.5. Then, by using the projection method for the time-dependent problem, the following result for the metastable motion of an interior one-spike solution for the shadow GM model was obtained in [44]:

+

+

+

Proposition 3.8. Let E + 0, and assume that either m = 2 and 1 < p 5 1 4 / N 1 or m = p 1 and 1 < p < pc, where p , is the critical Sobolev exponent. Then, a one-spike solution f o r the shadow GM model (3.25) with T = 0 is given asymptotically by a(x, t ) hq/(P-l)w ( E - ~ I x - xo(t)l),where x:o(t)satisfies the differential equation

+

+

N

Here r = 1x - 201, ? = ( x - x o ) / r , f i is the unit outward normal to dR, and c is defined in (1.7b). Next, assume that there is a unique point x , on dR, where r is minimized. Then, the spike moves exponentially slowly in a straight line towards x, and the distance rm (t )E 12, - xo(t)l satisfies .

In terms of the principal radii of curvature Ri, i = 1 , . . . ,N 1 of dR at x,, the function H(r,) is defined by H(r,) = -1/2

(1-2)

.+&)

-1/2

.

138

This result was first derived formally in Proposition 2 and Corollary 2 of [44] and was later proved rigorously in [14]. Metastability will also occur for other ( p , q , m, s) whenever we can guarantee that for any nonzero eigenvalue of (3.28) we have Re(X) < 0 (see Question 3.2 above). This analysis shows that an interior one-spike solution to the shadow GM niodel with T = 0 is ultimately unstable, and the spike will drift exponentially slowly towards the closest point on the boundary. An open problem concerns how the spike attaches to the boundary of the domain.

Question 3.5. Analyze the time-dependent motion of a spike when dist(z0,aR) = O ( E ) .How does a spike attach itself to the boundary?

We remark that if we were to change the boundary conditions from Neumann to the Robin condition Edna+Ka=O,

z ~ d R ,

(3.31)

where &a is the outward normal derivative, then from Eq. (2.58) of Sec. 2.5 we would expect that the exponentially small eigenvalues of (3.27) will all be negative when ri > 1 (see Proposition 2.1 for the analogous formula in one-dimension). Therefore, when K > 1 an interior one-spike equilibrium solution will be stable. This leads to the next question.

Question 3.6. Consider (3.25) (with T = 0) and with the Robin condition (3.31) for a with k > 1. Prove that an interior one-spike equilibrium solution is stable, and that a one-spike solution drifts exponentially slowly towards the point in dR that maximizes the distance to the boundary.

In [45] a formal asymptotic analysis was done for (3.25) when T = 0 to derive an equation of motion for a spike on the boundary dR of a domain. Since the spike is localized, to leading order we have a spike on the boundary of a half-space. In view of the Neumann boundary conditions, the stability of the spike profile on an O(1) time-scale is again determined by the infiniteline nonlocal eigenvalue problem (3.28). The following result was given in Proposition 2.1 of [45]: Proposition 3.9. Let E -+ 0 and assume that either m = 2 and 1 < p 5 3, or m = p + 1. Then, the motion of a spike for (3.25) that is confined to the

139

smooth boundary of a two-dimensional evolves according to (3.32a)

(3.32b)

Here w ( p ) satisfies (1.7) when N = 2, Q is the distance from x E R to dR, and s is the corresponding orthogonal coordinate, which measures arclength along dR when II = 0 . I n addition, K is the curvature of dR, taken with the sign convention that K > 0 for a circle. This result shows that the speed of the spike is O ( E ~and ) , that stable equilibrium points correspond to points on the boundary where K has local maxima. An analogous result for the spike motion on the boundary of a three-dimensional domain is given in Proposition 3.1 of [45]. For the full GM model (1.1) it was proved in [18) that there is an equilibrium boundary spike solution that concentrates at a local maximum of the curvature of dR whenever the inhibitor diffusivity D in (l.lb) is sufficiently large. Therefore, for equilibrium boundary spike solutions, the shadow GM model closely predicts behavior in the full GM model for D large. The result (3.32b) predicts that a spike is stationary on a flat segment of the boundary where K’ = 0. In this case, as was shown in [45], the motion of a spike is exponentially slow and is determined by the local behavior of the boundary at the ends of the flat segment. Consider the two-dimensional case where z = (x,y), and suppose that the spike is located on the straightline boundary segment joining the points ( x ~ , 0and ) ( z R , ~ )as shown in Fig. 5. The spike is centered at xo = (6,O) where X L < E < ZR. We decompose d R as OR = 80, u dR, where do, is the straight-line segment of the boundary and do, denotes the remaining curved part of 80. The distance between the spike and OR, is assumed to be a minimum at either of the two corners ( x ~ , 0or) ( z R , ~ ) . The local behavior of R, near the corner points is critical to determining the motion. Near these corner points, we assume the local behavior

- K L ( Z L - x)ar., as x -+

( Z L , ~ ;)

y = +L(z),

+L(x)

;

y = +R(z),

+ i ( x ) N K R ( X- Z

(ZR, 0)

N

,

R ) ~ as ~

51,

x +~

(3.33a)

2 (3.33b) ,

where Q L > 0 and CYR> 0. When Q L = Q R = 1, K L and K R are proportional to the curvature of dR, at the left and right corners, respectively.

140

(