Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

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Chapter 5: Periodic Solutions of Nonlinear Wave Equations

Specifying (n, k) = (1, 1) and (n, k) = (1, -1), substitution of the previous

equation in (5.102) yields for the Q-equations

2 2

+ 0(5) = 0

11 - \2 + p +

4(1-,\2)+ p - q

1 - A2 + p + 62,14

9(1-- P - _

+ 0(53) = 0.

4A -p

(5.104)

Assume p > 0, p # 4/3 fixed. For A2 = 1 + p + O(52), one may then solve

(5.104) for p = p(A), q = q(A). In order to be able to solve the P-equations

by Newton's method, a sequence of conditions on p, q, A are imposed that

eventually will restrict A down to some Cantor-like set C of positive measure,

such that for A E C one gets a solution of (5.93) of the form (5.97) with

p = p(A), q = q(A) in (5.98).

We now turn our attention to the P-equation. The linearization of (5.102)

at a given approximative solution y yields the linearized operator

T = (-k2A2 + n2 + p)Il + 25[S1

, o

(kA)1].

(5.105)

What follows is a bit easier in the present context of time-periodic solu-

tions than in the general quasiperiodic case, due to the fact that here, for

singular sites, necessarily ski - InI.

Use (5.101) to write

-k 2,\2 + n2 + p = (1 - k2)p + n2 - k2 + O(k262).

(5.106)

Since we excluded the resonant cases (n, k) = (1, 1) and (n, k) = (1, -1)

(n > 0) and p is assumed to fulfill typical Diophantine conditions, no singular

sites appear in the region Int + Iki < Mo = Mo(5) = 6-c'. Applying the

resolvent identity, the control of T-1 may then be achieved from that of Ti',

where A is a box of the form

A+= 1L<n< 11 L]

[

10AL<k<

12 L]

(5.107)

1110LIX[

L<-k<1012A]

(5.108)

with L > io MO (5), say. Assume (n, k) restricted to a box A+, say. Instead

of considering T, one may invert

T,_ -k2A2+n2+p

Il + 2SS

(5 109)

Bourgain

restricted to A+ (this operator is not self-adjoint since it is the sum of a

diagonal and an antisymmetric matrix). Defining the set of singular sites I,

of T' as the pairs (n, k) E A+ such that (c2 < cl sufficiently small)

I-k2 A2+n2+pl

<E",

we get, since k > M,

(5.110)

Ika - nI < 5". (5.111)

This permits us to conclude a separation of the singular sites by &-e3, for

some c3 > 0.

Indeed, if (nl, kl) E I (nzi kz) c I and Ikl - k2I < b, it

follows from (5.101) and (5.111) that

II (kl - k2) 1 + plI < 36". (5.112)

Hence, since p is assumed Diophantine,

Ik1 - kzI `° < II(ki - k2)2p1I < 10b°zIkl - k21,

(5.113)

Ik1 -

k2$

-c2/I+C > 8-C3

implying the previous statement about separation.

Our aim is to get for suitable restrictions of the parameters p, q, A again

good (polynomial) bounds on the inverse (TT+)-1 and off-diagonal decay.

This is achieved by an inductive process considering restrictions of the oper-

ator to boxes Q C A+ of increasing size scale, much in the spirit of [1], [3],

and [8]. These boxes will be obtained by restriction of the k-index only, thus

IL<n< 10L]

xJ,

(5.114)

where J is an interval. in [-!-L L < k <

-L] of a certain size o(L). The

multiscale analysis of the inverse of

-k2,\2kan2

II + SSIQ

(5.115)

will relate to size scales of J. Observe that in (5.115) k, n - L.

Restricting T' to the complement of I,, the inverse is controlled by a

Neumann series.

Chapter 5: Periodic Solutions of Nonlinear Wave Equations

Next we consider TT = T`IQ where Q is a box containing at most one

singular site and later on restrictions to boxes of increasing size order. This

induction process is mainly based on the separation properties of "singular"

boxes, meaning essentially that, for Q1, Q2 boxes of same size not contained

in the doubling of each other, say, large

II(T4,)-1II and

II(T4s)-1II

may only

occur when Ql and Q2 are sufficiently separated. These separation properties

are then used together with the resolvent identity to establish off-diagonal

decay of the inverse.

In view of (5.101), we introduce a new parameter r with variation range

0(1), letting

A2=1+p+b2r. (5.116)

Thus from (5.104) and (5.116) we have

ar N

b2, ap

= 0(1),

Ilq

= 0(1) (5.117)

OIT

(after expressing p, q in A from (5.104)). Parameter restrictions are eventually

expressed in terms of r, which admissible values will be restricted to a Cantor-

type set.

In order to set up the multiscale induction process, we introduce an extra

parameter a, Jul < a and consider the operators

TZ -(kA+o)2+n2+pI+bS(Q

(5.118)

kA+Q

The reason for this is the fact that

T°+koa

T°

o - - Q

(5.119)

and hence a discussion of (TQo)-1, Qo = [L < n < 1L] x Jo, Jo fixed, in

the full parameter range will also apply to k-translates Q of Qo. As in [1]

and [3], we will control

(TQo)-1 by

a system of monic polynomials in or with

smooth coefficients in A, p, q. In fact, these polynomials will be here of degree

1, which simplifies matters a bit. In what follows, we briefly point out the

main ideas.

From the considerations (5.110)-(5.113), it follows that, if Jo is sufficiently

short, say < b-`3, there is at most one singular site (no, ko) for TQo (fixing

the parameters). Thus

(kA+Q)2+n2+p

b f

or (n,

o}.

) E Qo\{no,

kA + o

(5.120)

Denote

Ql = Q + koa - no

(5.121)

Bourgain

assuming thus

lo1I < 812

(5.122)

Rewrite the diagonal of TQo as

[no+n+(k-ko)A+al][(n-no)+(ko-k)A-o1]+p

no +(k-ko)A+o1

(5.123)

_ [1 +

(1 + o(1))][n - no) - (k - ko)A - 01], (5.124)

710

where the o(1) also refers to o1 and A-derivatives and - 0 for Inol -i oo.

Write

T_ Tea\{no,ko}

-g-

St -(2*b+oj)

no+al

(5.125)

where, by (5.120), (TQo\{no,ko})-1 is well controlled.

To control the inverse of (TQo)-1, we need thus consider the reciprocal of

the expression

-(2no+oi)o1+P

2 0 1

n0+01

+(1 ((TQ0\{n0,ko})-CCS,ttS)

-201 +o(j)+S2 ((To\{no,ko})-1S, t)

-201 + W(o1 i A, R q)

considered as a function of o1i A, p, q. Thus

o1+P

i%1V = 0(1),

(5.126)

aavl N 82,

(5.127)

1,9,Vl, lagwvl N 62.

(5.128)

(In (5.127) and (5.128), an estimate of the form 62-2c2 is clear from

(5.120); the stated bound results from a slightly more careful analysis of

the last term in (5.126) invoking decay considerations.)

Chapter 5: Periodic Solutions of Nonlinear Wave Equations

From the implicit function theorem, one may then replace (5.126) by

al - 'b(A, p, q), (5.129)

where z/, still satisfies (5.127) and (5.128).

Observe that, from (5.121), if we let a = 0,

aa,

Nko - L>Mo. (5.130)

Hence, considering A, p, q as a function of T, weI have

[koa - no - ii(A,p, q)] > M082

(5.131)

by invoking (5.117).

In order to obtain separation of "singular" matrices TT, and TQ2 with

Qi = Qo + ki

(i=1,2)

(5.132)

and

Ik, - k21 >> IQoI,

(5.133)

one has to avoid simultaneous almost vanishing of two expressions

ai -

V51 (A, p, q)

0, (5.134)

a2 - 02(A, p, q)

0, (5.135)

where

ai = a + ko,iA - no,,,

Iki - ko,il < IQoI

(5.136)

If (5.134), (5.135), and (5.136) hold, we have (eliminating a)

(ko,l - ko,2)A - no,l + no,2 - zlii (A, p, q) + 02(A, p, q)

0. (5.137)

Hence

II(ko,i - k0,2)A-Vi(A,p,q)+V,2(A,p,q)II

0, (5.138)

where

Iko,i - ko,21 >> IQoI

(5.139)

Bourgain 89

To avoid this, we use the fact that

[(ko,l - ko,2).\ -'1(A,p, q) + 2(',p, q)] '

ko,1 - ko,2. (5.140)

Hence

[(ko,i - ko,2)A - 01 (A, p, q) + 2(A, p, q)]

> IQoI52

(5.141)

The preceding is the first step in the multiscale analysis. For the continu-

ation of the process, we refer the reader to [1] and [3]. This permits us for

the restrictions TQ, Q c A±, to get estimates on the inverse that are pow-

erlike in IQI with exponential off-diagonal decay; similarly for the inverse of

restrictions of T as needed in the Newton scheme. For details, the reader is

referred to [1] and [3].

The result of the preceding is the following.

Theorem 5.2. Consider a NLW equation

yet - yxx + py + (yt)2 = 0

(5.142)

with periodic boundary conditions, where p > 0 is a typical number. Then

for b sufficiently small taken in a Cantor set of positive measure, (5.139) has

a time-periodic solution of the form (5.97)-(5.99)

y(x, t)* = p cos(x + at) + q cos(x - .\t)

+ En>o,(n,k)#(1,1),(1,-1)

y(n, k)

cos(nx + At),

where

(5.143)

p = p(5), q = q(5) = O(5) and A2 = 1 + p + 0(52)

(5.144)

and the last term in (5.140) is 0(52).

5.3

Some Remarks on Frequencies and

Parameters

Consider a perturbed Hamiltonian system

8Ho

+ 6

OH1

(5.145)

iq=

ag,

Chapter 5: Periodic Solutions of Nonlinear Wave Equations

where q = {qn}n 1, N finite or N = oo, q,, c C are the pairs of conjugate

variables. Let 1 < b < N, 0 an interval in Rb, a E 0 a b-parameter, and

qoa)

qo 1(t)

(5.146)

a family of quasiperiodic solutions of

qoi

147)

(54

with frequency vector .A (a) _ (AI(a), ... , J1b(a)). Assume the nondegeneracy

condition

det I a2 I # 0

\ 8ak / 1< j,k <n

(5.148)

is satisfied.

When b = N, the problem of persistency of some of these tori for the

perturbed equation (5.145) is treated in the classical KAM theory. In this

case, invariant tori for (5.145) are obtained for (perturbed) frequency vectors

A' satisfying some Diophantine condition, of the form

I(A',k)I > clkl-r for all k E Zb\{0},

(5.149)

where c > 0, r > b - 1 are some constants.

Our interest goes to the case N > b. For b = 1, N finite, the discussion

below will not apply since there are no small divisors. However, in the PDE

context (N = oo) the problems which we recall next appear as well for b = 1.

Take for simplicity Ho of the form

Ho(q,4) _ Ean(a)Ignl2+E{ln(a)Jq.J'

(5.150)

n=1

n>b

corresponding in (5.147) to a parameter dependent linear equation.

For b < N, the persistency problem, which we call "Melnikov problem"

has been studied by various authors, including [6], [9], [10] and also in earlier

works of J. Moser [12, 13] (N < oo here). The method used in those works is

the standard Hamiltonian procedure. The nonresonance conditions required

are of the form

J(A', k) + V, t) I > c(JkJ + IiI)`

(5.151)

for (k, t) E Z "\{0}, It < 2.

This condition is more restrictive than (5.149) since the normal frequen-

cies {µ;a} are also involved. Observe that, in the context of (5.145), if we fix

H1 but let a vary, we have

A' = A'(a, e) and p' = µ'(a, e)

. (5.152)

Bourgain

Thus, if we choose a =: (al,

... ,

ab) such that

A' = A'(a, e) = Ao,

(5.153)

where A0 is fixed, satisfying (5.149), say, (5.151) is not necessarily satisfied

when b < N. As observed by J. Moser, to fulfill (5.151) requires more

parameters. Alternatively, Eliasson [7] weakens (5.153) to the property

A' = tAo for some t E R

(5.154)

(thus the tangential frequency is parallel to a given vector Ao).

Essentially speaking, if b parameters are available, (5.154) corresponds to

b-1 conditions and the remaining degree of freedom is used to ensure (5.151).

In conclusion, the set of "admissible" tangential frequencies for invariant tori

of (5.145) depends on e.

In the case N = oc, quasiperiodic solutions of Hamiltonian perturbations

of linear or integrable PDE's where studied by S. Kuksin [9, 10] and C. Wayne

[14], based again on the KAM methods. In particular, condition (5.151) is

needed.

The Lyapounov-Schmidt technique applied in previous sections, origi-

nating from [5] and developed further by the author, is less restrictive since

multiplicities in the normal frequencies are not excluded. This method allows

one to understand in particular how tori with positive Lyapounov exponents

appear in perturbations of integrable systems with only elliptic tori. Roughly

speaking, the relevant nonresonant expressions here are of the form

(A', k) + (p', f) with (k, B) E ZN\{O}, jti < 1. (5.155)

However (5.155) still involves the normal frequencies and previous comments

apply as well.

There are a number of natural questions one may formulate here. For

instance, if Eliasson's result [6] with parallel tangential frequency vector may

be obtained in the infinite-dimensional (PDE) setting (either following KAM

or Lyapounov-Schmidt procedure) or whether (5.153) may be ensured if extra

parameters available. Finally, the preceding discussion only comments on

the methods and does not claim failure of existence of invariant tori if the

nonresonance conditions are violated. One may, however, expect to prove

nonexistence of certain families of invariant tori with given frequency vector

A, differentiable in the perturbation parameters, since that statement relates

directly to the linearized equation.

In the remainder of this section, we give an example of such smooth fam-

ilies with Diophantine frequency A in the context of time periodic solutions

of a NLW equation with an extra parameter. Consider an equation

ytt - yaz + py + y3 + e f (X, y) = 0,

(5.156)

Chapter 5: Periodic Solutions of Nonlinear Wave Equations

where p will be a parameter. Assume for simplicity that f is a sum of the

form

f (x, y) = bi(x)yj (5.157)

f>1

with (bb} even trigonometric polynomials in x (real-analytic assumptions

should work as well). Thus according to (5.145)

Ho(y, v) =

f [(Y)2

+ ZY' +

4y4 +

2v2] , (5.158)

Hl (y, v) = f

F(x, y)

(8OF = f), (5.159)

with (y, v = y) as canonical variables.

Our parameters a will be extracted from the nonlinearity y3 by amplitude-

frequency modulation, and we let p in (5.156) be a second parameter.

For c > 0, 6 > 0, denote by A,,6 the set of frequencies A satisfying

following Diophantine conditions

IIkAII = min Ik) - nj >

clkl-'-6 for all k

E Z\{0}

(5.160)

and

I -(k2-1)A2+n2-11>Clkl-6 for allkEZ\{1,-1}.

(5.161)

(CA )

0 f

ll d

ear

y mes

,,6

or a >

We prove the following

Theorem 5.3. Let A E A,,6 (6 small enough). For sufficiently small e, there

are smooth functions of e

p = p(e) , a = a(e)

(5.162)

such that (5.156) has a A-periodic solution in time, of the form

y,(x, t) = a cos x cos At +

yE(n, k) cos nx cos kAt

(5.163)

n,k>O,(n,k)$(1,1)

depending smoothly on e and such that

e-(InI+IkU`I1/e(n,

k)I = 0(e + 1a13)

(5.164)

(n,k)#(1,1)

(a is small).

Bourgain

Note that the representation (5.163) is compatible with the assumption

that (5.157) is even in x. This result is in the spirit of some results from [12]

and [13].

We construct (5.163) following the Lyapounov-Schmidt scheme from [5],

perturbing off from the linear equation

ytt-yza+Py=0

(5.165)

with periodic solution

yo (x, t) = a cos x cos Aot

(5.166)

and

Ao = \o(P) =

+P .

(5.167)

The Q-equation is obtained by substituting (5.163) in (5.156) and projecting

on the (1, 1) mode. Thus we get

(-A2+1+p)a+y3(1,1)+ef(x,y)(1,1) = 0,

(5.168)

hence of the form

-A2+1+p+ 9 a2+O(a3)+O(e) = 0.

(5.169)

The system of P-equations

(-A2k2 + n2 + p)y(n, k) + y3(n, k) + of (x, y)(n, k) = 0,

(5.170)

(n, k) E Z+ x Z+\{1,1 }, is solved by the Newton iteration procedure.

At stage r, an approximate solution y,.

is obtained and the linearized

equation at yT is given by

T=D+S, (5.171)

where D is diagonal with diagonal elements

Dn k = -A2k2 + n2 + p + Q(0, 0) + e8y f (0, 0)

(5.172)

and S with 0-diagonal, given by the Toeplitz operator

S = Sm-m(oo)

(5.173)

and

0 = 3yf + e8y f (x, y,.)

(5.174)