Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions

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Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions

COHEN, David, HAIRER, Ernst, LUBICH, Christian

Abstract

A modulated Fourier expansion in time is used to show long-time near-conser-vation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.

COHEN, David, HAIRER, Ernst, LUBICH, Christian. Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Archive for Rational Mechanics and Analysis , 2008, vol. 187, no. 2, p. 341-368

DOI : 10.1007/s00205-007-0095-z

Available at:

http://archive-ouverte.unige.ch/unige:5204

Disclaimer: layout of this document may differ from the published version.

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(will be inserted by the editor)

Long-time analysis of

nonlinearly perturbed wave equations via modulated Fourier expansions

David Cohen, Ernst Hairer, Christian Lubich

Abstract

A modulated Fourier expansion in time is used to show long-time near- conservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.

Key words. semilinear wave equation, nonlinear Klein–Gordon equation, action preservation, long-time regularity, modulated Fourier expansion

1. Introduction

We consider the one-dimensional semilinear wave equation (nonlinear Klein-Gordon equation)

utt−uxx+ρu+g(u) = 0 (1)

for t > 0 and −π ≤ x ≤ π subject to periodic boundary conditions. We assume ρ > 0 and a nonlinearity g that is a smooth real function with g(0) =g⁰(0) = 0. We take small initial data: in appropriate Sobolev norms, the initial datau(·,0) and∂tu(·,0) is bounded by a small parameterε, but is not restricted otherwise. We are interested in studying the behaviour of the solutions over long times t ≤ ε^−N, with fixed, but arbitrary positive integerN. Under a non-resonance condition that restricts the possible val- ues of ρto a set of full measure, we show that for each Fourier mode, the harmonic action remains nearly constant over such long times, as does the Sobolev-type norm specifying the smallness of the initial data. The result slightly refines previous results by Bambusi [1] and Bourgain [5], using en- tirely different techniques.

The main novelty in the present paper lies in the technique of proof via amodulated Fourier expansion in time. This is a multiscale expansion that represents the solution as an asymptotic series of products of exponentials e^iωjt (with ωj the frequencies of the linear equation) multiplied with co- efficient functions that vary slowly in time. This approach was first used for showing long-time almost-conservation properties in [11], in that case of numerical methods for highly oscillatory Hamiltonian ordinary differential equations; also see [12, Ch. XIII] and further references therein. A modu- lated Fourier expansion appears similarly, and independently, in the work by Guzzo and Benettin [10] on the spectral formulation of the Nekhoroshev theorem for quasi-integrable Hamiltonian systems. In the context of wave equations, the expansion constructed here can be viewed as an extension to higher approximation order of a nonlinear geometric optics expansion given by Joly, M´etivier, and Rauch [13]. Multiscale expansions and modulation equations have certainly been used in various types and for various purposes in many places, also with nonlinear wave equations; see, e.g., Whitham [16], Kalyakin [14], Kirrmann, Schneider, and Mielke [15], Craig and Wayne [8].

Unlike all these works, we here construct a two-scale expansion to arbitrary order inεand use the Lagrangian/Hamiltonian structure of the modulation equations to infer long-time near-conservation and regularity properties over timesε^−N, beyond the time of validity ε⁻¹ of the expansion.

In Section 2 we describe the technical framework and state the result on the long-time near-conservation of harmonic actions (Theorem 1). Section 3 gives the construction of the modulated Fourier expansion and proves the necessary bounds of its coefficients and of the remainder term, which are collected in Theorem 2. The expansion works with all frequencies in the system, without a cutoff of high frequencies. While Section 3 is the technical core of this paper, its conceptual heart is in Section 4. There, it is shown that the system determining the modulation functions has a Hamiltonian structure and a remarkable invariance property, which yields the existence of almost-invariants close to the harmonic actions (Theorems 3 and 4).

Though the modulated Fourier expansion is constructed only as a short- time expansion (over time scaleε⁻¹), its almost-invariants can be patched together over very many short time intervals, which finally gives the long- time near-conservation of actions over times ε^−N with N >1 as stated in Section 2.

The approach to the long-time analysis of (1) via modulated Fourier expansions doesnotuse nonlinear coordinate transforms to a normal form, as is done in Bambusi [1] (see also [2–4,6] and references therein) and as is typical in Hamiltonian perturbation theory. While normal form theory uses coordinate transforms to take the equation to a simpler form from which essential properties can be read off, the present approach can be viewed as instead embedding the original equation in a large system of modulation equations from which the desired properties can be read off.

We consider equation (1) only with periodic boundary conditions, but it appears that the problem with Dirichlet boundary conditions, as studied

in [1,5], can be treated in the same way. Less general than [1], we do not treat problems with an additional dependence on x in ρ and g, though such an extension could be done without pain. As in these previous works, an extension of the results to problems in more than one space dimension over time scalesε^−N withN > 1 does not seem possible with the present techniques, mainly due to problems with small denominators. See, however, Delort and Szeftel [9] for existence results over timeε⁻²for nonlinear wave equations with periodic boundary conditions in higher dimension. Moreover, Bambusi, Delort, Gr´ebert and Szeftel [4] deal with the same equation on manifolds (e.g. spheres) of arbitrary dimension.

Corresponding to the authors’ research background, the present work was originally motivated by numerical analysis, with the aim of understand- ing the long-time behaviour of discretization schemes for the nonlinear wave equation (1). Since the approach via modulated Fourier expansions does not use nonlinear coordinate transforms, it turns out to be applicable also to numerical discretizations of (1), as is shown in a companion paper to the present article [7].

2. Preparation and statement of result

In this section we describe basic concepts, introduce notation, formulate assumptions, and state the result on the long-time near-conservation of actions.

2.1. Modulated Fourier expansion

The spatially 2π-periodic solutions to the linear wave equation ∂_t²u−

∂_x²u+ρu = 0 are superpositions of plane waves e^±iωjt±ijx, where j is an arbitrary integer and

ωj=p ρ+j²

are the frequencies of the equation. If the nonlinearityg is evaluated at a superposition of plane waves, its Taylor expansion involves mixed products of such waves. This can be taken as a motivation to look for an approx- imation to the solution u(x, t) of the nonlinear problem in the form of a modulated Fourier expansion, that is, a linear combination of products of plane waves with coefficient functions that change slowly in time, or more precisely, their derivatives with respect to the slow timeτ=εt are bounded independently ofε:

u(x, t)≈eu(x, t) = X

kkk≤K

z^k(x, εt) e^i(k·ω)t= X

kkk≤K

X∞ j=−∞

z_j^k(εt) e^{i(k·ω)t+ijx}. (2) Here, the sum is over all

k= (k`)`≥0 with integersk`and kkk:=X

`≥0

|k`| ≤K

(at mostKof thek` are non-zero) and we write k·ω=X

`≥0

k`ω`.

ForK= 2N, we will obtain an expansion (2) with an approximation error of size O(ε^N+1) in the same norm in which the initial data is assumed to be bounded byε, uniformly over timesO(ε⁻¹).

In the construction, a special role is played by the modulation functions z_j^k fork=±hji, with the notation (Kronecker delta)

hji:= (δ|j|,`)`≥0.

The function z_j^±hji is multiplied with e^±iωjt+ijx in (2). The z_j^±hji will be determined from first-order differential equations, whereas the otherz^k_j are obtained from equations of the form ω²_j −(k·ω)²

z_j^k = . . . , where we need to divide by ωj− |k·ω|. If this denominator is too small in absolute value (less than ε^1/2, say), then this corresponds to a situation where we cannot safely distinguish the exponentials e^±iωjt and e^±i(k·ω)t and we just setz_j^k= 0.

2.2. Non-resonance condition

The effect of ignoring contributions from near-resonant indices (j,k) for which|ωj±k·ω|< ε^1/2(+ or−), should not spoil theO(ε^N+1) remainder term in the modulated Fourier expansion. This requirement is fulfilled under anon-resonance condition. With the abbreviations

|k|= (|k`|)`≥0 and ω^σ|k|=Y

`≥0

ω<sub>`</sub><sup>σ|k`|</sup> (3) and the set of near-resonant indices

Rε={(j,k) :j ∈Zandk6=±hji,kkk ≤2N with|ωj±k·ω|< ε^1/2}, (4) the non-resonance condition can be formulated as follows: there areσ >0 and a constantC0 such that

sup

(j,k)∈Rε

ω_j^σ

ω^σ|k|ε^kkk/2≤C0ε^N. (5) For N = 1, this condition is always satisfied for arbitraryσ ≥0 and ρin (1). ForN >1, it imposes a restriction on the choice ofρ, and the possible values ofσdepend onN. The condition requires that a near-resonance can only appear with at least two large frequencies among the ω` with k`6= 0 (counted with their multiplicity|k`|).

As we show next, condition (5) is implied, for sufficiently largeσ, by the non-resonance condition of Bambusi [1], which reads as follows: for every

positive integer r, there exist α = α(r) > 0 and c > 0 such that for all combinations of signs,

|ωj±ωk±ω`1±. . .±ω`r| ≥c L^−α for j≥k≥L=`1≥. . .≥`r≥0, (6) provided that the sum does not vanish because the terms cancel pairwise.

In [1] it is shown that for almost all (w.r.t. Lebesgue measure)ρin a fixed interval of positive numbers there is a c >0 such that condition (6) holds withα= 16r⁵. It is also noted in [1] that an analogous condition is typically not satisfied for wave equations in spatial dimension greater than 1.

Lemma 1. Under Bambusi’s non-resonance condition (6), the bound (5) holds with σ= maxr+1<2N(2N−r−1)α(r).

Proof. Consider (j,k)∈ Rε, so that|ωj− |k·ω||< ε^1/2. Forkwithkkk= r+1, we writek·ω=±ωm±ω`1±. . .±ω`r withm≥L=`1≥. . .≥`r≥0.

We then have ωj

ω^|k| ≤ |k·ω|+ε^1/2

ω^|k| ≤ ωm+ω`1+· · ·+ω`r+ε^1/2 ωmω`1. . . ω`r

≤C L where the constantC depends on a lower bound ofρ.

Now, under condition (6), a near-resonance|ωj±k·ω|< ε^1/2 can only appear withcL^−α< ε^1/2, i.e.,L⁻¹< c^−1/αε^1/(2α). We then have

ω_j^σ

ω^σ|k| ≤ C^σ

L^σ ≤C0ε^σ/(2α)

with C0 = (C/c^1/α)^σ. Ifσis chosen so large that σ/(2α)≥N −¹₂(r+ 1), i.e.,σ≥(2N−r−1)α, then we obtain the bound (5). ut

With Bambusi’s valueα(r) = 16r⁵, the lemma yieldsσ= 2⁹already for N = 2. (The corresponding quantity in [1] iss∗= 4M α(2M) forM =N+2, which for N = 2 results in s∗ = 2¹⁹.) However, it should be noted that condition (5) may actually be satisfied with a much smaller exponent σ.

This is suggested by testing (5) numerically for various values ofε,ρ, and N.

2.3. Functional-analytic setting: Sobolev algebras

For a 2π-periodic function v ∈L²(T) (with the circle T=R/2πZ), we denote by (vj)j∈Zthe sequence of Fourier coefficients ofv(x) =P∞

−∞vje^ijx. We will work with functions (or coefficient sequences) for which the weighted

`² norm

kvks= X^∞

j=−∞

ω_j^2s|vj|²1/2

is finite. We denote, fors≥0, the Sobolev space

H^s={v∈L²(T) :kvks<∞}={v: (−∂_x²+ρ)^s/2v∈L²}.

Fors > ¹₂, we haveH^s⊂C(T) andH^sis a normed algebra:

kvwks≤Cskvkskwks. (7) It is convenient to rescale the norm such thatCs= 1.

2.4. Condition of small initial data

We assume that the initial position and velocity have small norms in H^s+1 andH^s, resp., for ans≥σ+ 1 withσof the non-resonance condition (5):

ku(·,0)k²_s+1+k∂tu(·,0)k²_s1/2

≤ε. (8)

This is equivalent to requiring X∞

j=−∞

ω^2s+1_j ωj

2 |uj(0)|²+ 1 2ωj

|∂tuj(0)|²

≤ ¹

2ε². (9)

2.5. Long-time near-conservation of harmonic actions Along every real solution u(x, t) =P∞

j=−∞uj(t) e^ijx to the linear wave equation∂_t²u−∂_x²u+ρu= 0, theactions(energy divided by frequency)

Ij(t) =ωj

2 |uj(t)|²+ 1 2ωj

|∂tuj(t)|²

remain constant in time. For real solutions as considered here, we have u_−j=ujand henceI_−j =Ij. For thenonlinearequation (1) with a smooth real nonlinearity satisfyingg(0) = g⁰(0) = 0, and under conditions (5) and (8), we will show that the actionsIj and in fact also their weighted sums

X∞ j=−∞

ω_j^2s+1Ij(t) = ¹

2ku(·, t)k²_s+1+¹

2k∂tu(·, t)k²_s,

remain constant up to small deviations over long times.

Theorem 1. Under the non-resonance condition (5) and assumption (8) on the initial data with s≥σ+ 1, the estimate

X∞

`=0

ω<sub>`</sub><sup>2s+1</sup>|I`(t)−I`(0)|

ε² ≤Cε for 0≤t≤ε^−N+1

holds with a constantC which depends ons,N, andC0, but not onεandt.

This result is closely related to results by Bambusi [1] and Bourgain [5].

In particular, Bambusi shows that, under the non-resonance condition (6) and with the same assumption on the initial data, there is the estimate

|I`(t)−I`(0)|/ε² ≤Cε ω_`^−2(s+1), which is close to the above bound. Theo- rem 1 implies, in particular, that the same norm that specifies the smallness condition on the initial data, remains nearly constant along the solution over long times: fort≤ε^−N+1,

ku(·, t)k²_s+1+k∂tu(·, t)k²_s=ku(·,0)k²_s+1+k∂tu(·,0)k²_s+O(ε³). This could also be obtained as an immediate consequence of the theory of [1]. Theorem 1 can be further interpreted as saying that the solution (u(t), ∂tu(t)) stays close to an infinite-dimensional torus in theH^s+1/2norm for long times. This improves slightly on [1], where such an estimate is obtained only in weaker norms.

We remark that forcomplexsolutions of (1) with a complex differentiable nonlinearityg, the statement of Theorem 1 remains valid with

I`(t) =ω`

2 u−`(t)u`(t) + 1

2ω`∂tu−`(t)∂tu`(t), with the same proof.

We emphasize that the main novelty of the present work is not in the result, but in the technique of proof via invariance properties of the system of equations that determine the coefficient functions in the modulated Fourier expansion (2). This approach is completely different from the techniques in [1,5] and turns out to be applicable also to numerical discretizations of (1), since it involves no transformations of coordinates.

2.6. Illustration of the near-conservation of actions

In this subsection we give numerical results that show long-time near- conservation of actions even in situations that are not covered by Theo- rem 1: for initial data that are not very smooth and, more remarkably, near-conservation of the actions corresponding to high frequencies even for initial data that are not small. At present we have no rigorous explanation of these phenomena.

We consider the nonlinear wave equation (1) withρ= 1 and nonlinearity g(u) = u², subject to periodic boundary conditions. As initial data we choose

u(x,0) =ε 1−x²

π² 2

, ∂tu(x,0) = 0 for −π≤x≤π. (10) The 2π-periodic extension ofu(x,0) has a jump in the third derivative, so that its Fourier coefficientsuj(0) decay likej⁻⁴. This function therefore lies inH^swiths≤3.

Figure 1 shows the first 32 functions I`(t) on the time interval [0,1000]

(they are computed numerically with high precision). We have chosen a

0 300 600 900 10⁻¹²

10⁻⁹ 10⁻⁶ 10⁻³

Fig. 1. Near-conservation of actions; the first 32 actions I_`(t) are plotted as functions of time.

large ε = 0.5, so that we are able to see oscillations at least in the low frequency modes. The higher the frequency, the better the relative error of the corresponding action is conserved. Forεsmaller than 0.1 only horizontal straight lines could be observed. Further experiments with this example have shown that the qualitative behaviour of Fig. 1 is insensitive with respect to the value ofρ, as long as it is not too small, and the good conservation holds on much longer time intervals.

3. The modulated Fourier expansion

Our principal tool for the long-time analysis of the nonlinearly perturbed wave equation is a short-time expansion constructed in this section.

3.1. Statement of result

We will prove the following result, where we use the abbreviation (3) and, fork= (k`)<sub>`≥0</sub>with integersk` andkkk=P

`|k`|, we set

[[k]] =





 1

2(kkk+ 1), k6= 0 3

2, k= 0.

(11)

Theorem 2. Consider the nonlinear wave equation (1) with frequencies ωj satisfying the non-resonance condition (5), and with small initial data bounded by (8) with s≥σ+ 1. Then, the solution u admits an expansion (2),

u(x, t) = X

kkk≤2N

z^k(x, εt) e^i(k·ω)t+r(x, t), (12)

where the remainder is bounded by

kr(·, t)ks+1+k∂tr(·, t)ks≤C1ε^N for 0≤t≤ε⁻¹. (13) On this time interval, the modulation functionsz^k are bounded by

X

kkk≤2N

ω^|k|

ε^[[k]]kz^k(·, εt)ks

2

≤C2. (14)

Bounds of the same type hold for any fixed number of derivatives ofz^k with respect to the slow time τ=εt. Moreover, the modulation functions satisfy z_−j^−k =z_j^k. The constantsC1 and C2 are independent of ε, but depend on N,s, onC0 of (5), and on bounds of derivatives of the nonlinearityg.

Apart from the relation z⁻_−j^k = z_j^k, the theorem and its proof remain unchanged for complex solutions of (1) with a complex differentiable non- linearity.

3.2. Formal modulation equations

Formally inserting the ansatz (2) into (1), equating terms with the same exponential e^{i(k·ω)t+ijx} and Taylor expansion ofglead to the condition

ω_j²−(k·ω)²

z_j^k+2iε(k·ω) ˙z_j^k + ε²z¨_j^k (15) +Fj

X

m

X

k¹+···+km=k

1

m!g^(m)(0)z^k1. . . z^km = 0. Here,Fjv=vj denotes thejth Fourier coefficient of a function v∈L²(T), and the dots (·) onz_j^k(τ) symbolize derivatives with respect toτ=εt. From this formal consideration, it becomes obvious that there will be three groups of modulation functions z_j^k: fork =±hji, the first term vanishes and the second term with the time derivative ˙z_j^k can be viewed as the dominant term. Fork6=±hji, the first term is dominant if|ωj±k·ω| ≥ε^1/2. Else, we simply set z^k_j ≡0 and we will use the non-resonance condition (5) to ensure that the defect in (15) is only of size O(ε^N+1) in an appropriate Sobolev-type norm.

In addition, the initial conditionsu(·,e 0) =u(·,0) and∂teu(·,0) =∂tu(·,0) need to be taken care of. They will yield the initial conditions for the func- tionsz^±hji_j :

X

k

z_j^k(0) =uj(0), X

k

i(k·ω)z_j^k(0) +εz˙_j^k(0)

=∂tuj(0). (16)

3.3. Reverse Picard iteration

We now turn to an iterative construction of the functionsz_j^k such that after 4N iteration steps, the defect in equations (15) and (16) is of size O(ε^N+1) in theH^snorm. The iteration procedure we employ can be viewed as a reverse Picard iteration on (15) and (16): indicating by [·]ⁿ that the nth iterate of all appearing variablesz_j^kis taken within the bracket, we set fork=±hji

±2iεωj

hz˙_j^±hjiin+1

=−h

ε²z¨_j^±hji+Fj

XN m=2

X

k1+···+km=±hji

g^(m)(0)

m! z^k1. . . z^kmin

and fork6=±hjiandj with|ωj±k·ω| ≥ε^1/2 we set ω²_j−(k·ω)²h

z^k_jin+1

=−h

2iε(k·ω) ˙z_j^k+岨z^k_j + Fj

XN m=2

X

k¹+···+km=k

1

m!g^(m)(0)z^k1. . . z^kmin

,

whereas we letz_j^k= 0 fork6=±hjiwith|ωj±k·ω|< ε^1/2. On the initial conditions we iterate by

hz_j^hji(0) +z^−hji_j (0)in+1

=h

uj(0)− X

k6=±hji

z_j^k(0)in

iωj

hz_j^hji(0)−z^−hji_j (0)in+1

=h

∂tuj(0)− X

k6=±hji

i(k·ω)z_j^k(0)−ε X

kkk≤K

˙ z_j^k(0)in

.

In all the above formulas, we tacitly assumekkk ≤K= 2N andkkⁱk ≤K.

In each iteration step, we thus have an initial value problem of first-order differential equations forz_j^±hji(j ∈Z) and algebraic equations forz_j^k with k6=±hji.

The starting iterates (n = 0) are chosen asz_j^k = 0 fork 6=±hji, and z_j^±hji(τ) =z_j^±hji(0) with z_j^±hji(0) determined from the above formula with right-hand sidesu(0) and∂tu(0).

For real initial data we haveu−j(0) =uj(0) and∂tu−j(0) =∂tuj(0), and we observe that the above iteration yields

z_−j^−kn

= z^k_jn

for all iteratesn and allj,kand hence gives real approximations (2).

3.4. Inequalities for the frequencies

We collect a few inequalities involving the frequencies ω`, which are needed later on. These inequalities only rely on the growth propertyω`∼` for large `, but do not depend on any diophantine relations between the frequencies.

Lemma 2. Fors >¹₂, X

kkk≤K

ω^−2s|k|≤CK,s <∞, (17) where we have used the short-hand notation (3). Fors > ¹₂ andm≥2, we have

sup

kkk≤K

X

k1+...+km=k

ω^{−2s(|k1|+···+|km|)}

ω^−2s|k| ≤Cm,K,s<∞, (18) where the sum is taken over (k¹, . . . ,k^m) satisfying kkⁱk ≤K. For s≥1, we further have

sup

kkk≤K

P

`≥0|k`|ω^2s+1_`

ω^2s|k|(1 +|k·ω|) ≤CK,s<∞. (19) Proof. We notice that

X

0<kkk≤K

ω^−2s|k|≤2 XK q=1

X∞

`=0

ω_`^−2s q

.

The termω^−2sq_{`1</sub> <sup>1</sup>· · · ω<sub>`}^−2sq_m ^m with 0≤`1< . . . < `m andq1+. . .+qm=q (qj >0) appears exactly 2^mtimes in the left-hand expression and _q1,...,q^q

m

times in P∞

`=0ω<sub>`</sub>^−2sq

(multinomial theorem). The estimate thus follows from the bound

2^m−1≤

q q1, . . . , qm

which is obtained by induction onm. The statement of the first inequality (17) is thus a consequence of the facts thatω`∼`andP

`≥1`^−2s<∞.

The second inequality (18) is proved as follows: wheneverk¹+. . .+k^m= k andkkⁱk ≤K, there exist q (with 0≤q≤mK) integers`1, . . . , `q ≥0 such that

|k¹|+· · ·+|k^m|=|k|+h`1i+· · ·+h`qi.

Conversely, for any choice of non-negative integers`1, . . . , `q withq≤mK, the number of (k¹, . . . ,k^m) satisfying k¹+. . .+k^m = k and the above equation, is bounded by a constantMm,K. Therefore,

X

k1+...+km=k

ω^{−2s(|k1|+···+|km|)}

ω^−2s|k| ≤Mm,K

XmK q=0

X

`1,...,`q≥0

ω^{−2s(h`1i+···+h`qi)}

=Mm,K mKX

q=0

X∞

`1=0

ω^−2s_`1 · · · X∞

`q=0

ω_`^−2s_q ≤Cm,K,s, which proves (18).

For the proof of (19) we split the set of kwith kkk ≤K into two sets:

for those k with |kL| = 1 and k` = 0 for all ` 6= L with ω` ≥ ω_L^1/2,

we have P

`≥0|k`|ω^2s+1_` ≤ ω^2s+1_L +Kω^s+1/2_L but ω^2s|k| ≥ cKω_L^2s with cK = min(1, ρ^2sK) and |k·ω| ≥ ωL−Kω^1/2_L , and hence the quotient of (19) is uniformly bounded on this subset ofk. On the complementary sub- set, we haveP

`≥0|k`|ω_`^2s+1 ≤Kω^2s+1_L for the largest integerL for which kL 6= 0, but here the product in the denominator is bounded from below as ω^2s|k|=Q

`≥0ω<sub>`</sub>^2s|k`|≥cK ω_L^1/22s

·ω_L^2s, and hence the quotient is uni- formly bounded on this subset fors≥1. This proves (19). ut

3.5. Rescaling and estimation of the nonlinear terms

Since we aim for (14), for the following analysis it is convenient to work with rescaled functions

c^k_j =ω^|k|

ε^[[k]] z_j^k, c^k(x) = X∞ j=−∞

c^k_je^ijx =ω^|k|

ε^[[k]] z^k(x) (20) where we use the notation (11) and (3). The superscriptskare in

K={k= (k`)`≥0with integersk` : kkk ≤K= 2N}, and we will work in the Hilbert space

H^s:= (H^s)^K={c= (c^k)^k∈K:c^k∈H^s} with norm k|ck|²_s=X

k∈K

kc^kk²_s=X

k∈K

X∞ j=−∞

ω_j^2s|c^k_j|².

We now express the nonlinearity in (15), v^k(z) =

XN m=2

g^(m)(0) m!

X

k1+···+km=k

z^k1. . . z^km,

withkkⁱk ≤K in the sum, in rescaled variables as the map f = (f^k)^k_∈K : H^s→H^sgiven by

f^k(c) = ω^|k| ε^[[k]]

XN m=2

g^(m)(0) m!

X

k¹+···+km=k

ε^{[[k1]]+···+[[km]]}

ω^{|k1|+···+|km|} c^k1. . . c^km. Using the triangle inequality, the inequality (PN

m=1am)² ≤ NPN m=1a²_m, and the Cauchy-Schwarz inequality, we obtain

k|f(c)k|²_s= X

kkk≤K

kf^k(c)k²_s

≤ X

kkk≤K

N XN m=2

g^(m)(0) m!

2 X

k¹+···+km=k

ε^{[[k1]]+···+[[km]]}

ε^[[k]]

ω^{−(|k1|+···+|km|)} ω^−|k|

2

× X

k1+···+km=k

kc^k1. . . c^kmk²_s.

Since H^s is a normed algebra, and since we have the bound (18) (with 1 in place of s there) and the obvious lower estimate [[k¹]] +· · ·+ [[k^m]] ≥

m−1

2 + [[k]], this is further estimated as X

kkk≤K

kf^k(c)k²_s

≤N XN m=2

g^(m)(0) m!

2

ε^m−1Cm,K,1

X

kkk≤K

X

k1+···+km=k

kc^k1k²_s. . .kc^kmk²_s

≤N XN m=2

g^(m)(0) m!

2

ε^m−1Cm,K,1

X

kkk≤K

kc^kk²_sm

=ε P(k|ck|²_s) (21)

where the polynomialP(µ) =NPN m=2

_g(m)(0) m!

2

Cm,K,1ε^m−2µ^mhas coef- ficients bounded independently ofε.

For k =±hji we note that if m ≥2 andk¹+· · ·+k^m =±hji, then necessarily [[k¹]] +· · ·+ [[k^m]]≥5/2. Hence, for the restriction to this case the bound improves to a factorε³ instead ofε:

X∞ j=−∞

kf^±hji(c)k²_s≤ε³P1(k|ck|²_s), (22)

whereP1is another polynomial with coefficients bounded independently of ε.

SinceH^sis a normed algebra and the mapf is an absolutely convergent sum of polynomials in the functionsc^k, we also obtain that f is arbitrarily differentiable with correspondingly bounded derivatives on bounded subsets ofH^s.

Instead of (20), we could also have worked with a different rescaling:

b

c^k_j =ω^s|k|

ε^[[k]] z^k_j , bc^k(x) = X∞ j=−∞

b

c^k_je^ijx= ω^s|k|

ε^[[k]] z^k(x), (23) considered in the spaceH¹= (H¹)^Kwith normk|bck|²₁=P

kkk≤Kkbc^kk²₁. For fb^kdefined in the same way asf^k above, but withω^s|k|in place ofω^|k|, we then have the bounds

X

kkk≤K

kfb^k(bc)k²₁≤εP(k|bck|²₁) X∞

j=−∞

kfb^±hji(bc)k²₁≤ε³P1(k|bck|²₁).

(24)

3.6. Abstract reformulation of the iteration

Forc= (c^k_j)∈H^swithc^k_j = 0 for allk6=±hjiwith|ωj±k·ω|< ε^1/2, we split the components ofccorresponding tok=±hjiandk6=±hjiand collect them ina= (a^k_j)∈H^sandb= (b^k_j)∈H^s, respectively:

a^k_j =c^k_j ifk=±hji, and 0 else

b^k_j =c^k_j if|ωj±k·ω| ≥ε^1/2, and 0 else. (25) We then have a+b=cand k|ak|²_s+k|bk|²_s=k|ck|²_s. We define the multi- plication operator onH^s,

(Ω⁻¹c)^k_j = 1

ωj+|k·ω|c^k_j for c∈H^s,

and note in particular that (Ω⁻¹c)^±hji_j =_2ω¹_jc^±hji_j . In terms ofaandb, the iteration of Subsection 3.3 written in the scaled variables (20) then becomes of the form

˙

a⁽ⁿ⁺¹⁾=Aa⁽ⁿ⁾+Ω⁻¹F(a⁽ⁿ⁾,b⁽ⁿ⁾)

b⁽ⁿ⁺¹⁾=Bb⁽ⁿ⁾+Ω⁻¹G(a⁽ⁿ⁾,b⁽ⁿ⁾), (26) with the linear differential operatorsAandB given by

(Aa)^±hji_j =± iε 2ωj

¨

a^±hji_j , (Bb)^k_j = −2iε(k·ω)

ω_j²−(k·ω)²b˙^k_j − ε²

ω_j²−(k·ω)² ¨b^k_j , forω_j− |k·ω|≥ε^1/2, and nonlinear mapsFand Ggiven by

F(a,b)±hji

j =±iε⁻¹f_j^±hji(a+b) G(a,b)^k

j =− ε^1/2

ωj− |k·ω|ε^−1/2fj^k(a+b).

In view of (21) – (22),FandGare arbitrarily differentiable maps, bounded inH^sbyO(ε^1/2) andO(1), respectively, with all derivatives bounded in the same way on bounded subsets ofH^s. The loss of a factorε^1/2in the bound forGresults from the condition|ωj±k·ω| ≥ε^1/2in (25). We further note the bounds

k|(Aa)(τ)k|_s≤Cεk|¨a(τ)k|_s,

k|(Bb)(τ)k|_s≤Cε^1/2k|b(τ)k|˙ _s+Cε^3/2k|b(τ)k|¨ _s. (27) The initial value fora⁽ⁿ⁺¹⁾is determined by an equation of the form

a⁽ⁿ⁺¹⁾(0) =v+Pb⁽ⁿ⁾(0) +Q a˙⁽ⁿ⁾(0) + ˙b⁽ⁿ⁾(0)

(28) where the nonzero componentsv_j^±hji ofvare given by

v_j^±hji= ωj

ε 1

2uj(0)±1

2(iωj)⁻¹∂tuj(0)

so that v is bounded in H^s by the assumption on the initial values, and with the operatorsP andQgiven by

(Pb)^±hji_j =−1 2ε

X

k6=±hji

ε^[[k]]

ω^|k|(ωj±k·ω)b^k_j (Qc)^±hji_j =±i

2 X

kkk≤K

ε^[[k]]

ω^|k|c^k_j ,

for which we have the bounds, using (17) with 1 in the role ofsthere, k|Pbk|_s≤Ck|Ωbk|_s, k|Qck|_s≤Cεk|ck|_s.

The starting iterate isa⁽⁰⁾=v and b⁽⁰⁾= 0.

3.7. Bounds of the modulation functions

The iteratesa⁽ⁿ⁾ andb⁽ⁿ⁾ and, by differentiation of the iteration equa- tions (26), also their derivatives with respect to the slow time τ =εt are thus bounded inH^sfor 0≤τ ≤1 andn≤4N: more precisely, the 4N-th iterates satisfy, with constants depending onN,

k|a^(4N)(0)k|_s≤C , k|Ωa˙^(4N)k|_s≤Cε^1/2, k|Ωb^(4N)k|_s≤C . (29) We also obtain the boundk|Ωb˙^(4N)k|_s≤Cand similarly for higher deriva- tives with respect toτ =εt. Forz_j^k=ε^[[k]]ω^−|k|c^k_j with (c^k_j) =c=c^(4N)= a^(4N)+b^(4N), the bounds foraand btogether yield the bound (14).

Refined estimates are obtained for components corresponding to the non- resonant setN ={(j,k) : ω²_j−(k·ω)²≥c}, wherec >0 is independent ofε. For indices in this set we gain the factorε^1/2in the estimate ofΩ⁻¹G, so that from the iteration (26) we obtain, withb=b^(4N),

X

(j,k)∈N

ω_j^2sb^k_j²1/2

≤Cε^1/2. (30)

In particularN contains all (j,k) withk=0, and those withk=±hj1i ± hj2i and j = j1 +j2 with all combinations of signs. Using (22), an even better bound is obtained forkkk= 1:

X

kkk=1

kΩb^kk²_s1/2

≤Cε^3/2. (31)

The bounds (29) implyk|c(τ)−a(0)k|_s+1 ≤C for c=c^(4N) and a = a^(4N), and hence give a bound of the expansion (2) in theH^s+1 norm:

ku(·, t)ke ²_s+1= X∞ j=−∞

ω_j^2(s+1) X

kkk≤K

z_j^k(εt)e^i(k·ω)t

2

≤ X∞ j=−∞

ω_j^2(s+1) ε ωj

|a^hji_j (0)|+|a^−hji_j (0)|

+ X

kkk≤K

ε^[[k]]

ω^|k||c^k_j(εt)−a^k_j(0)|2

≤4ε²k|a(0)k|²_s+CK,1ε² X∞ j=−∞

ω^2(s+1)_j X

kkk≤K

|c^k_j(εt)−a^k_j(0)|²

= 4ε²k|a(0)k|²_s+CK,1ε²k|c(εt)−a(0)k|²_s+1,

where we noteda^k_j = 0 fork6=±hjiand where we used the Cauchy-Schwarz inequality and (17) in the last inequality. So we have

ku(·, t)ke s+1≤Cε for t≤ε⁻¹. (32) With the alternative scaling (23) we obtain, again forτ=εt≤1,

k|ba^(4N)(0)k|₁≤C , k|Ωba˙^(4N)k|₁≤Cε^1/2, k|Ωbb^(4N)k|₁≤C . (33) The bounds for ba follow trivially from (29) and k|bak|₁ = k|ak|_s, those for b

b are obtained from the rescaled iteration (26) for bb⁽ⁿ⁾ and the bounds (24), without consideration of the starting values forba⁽ⁿ⁾. We also obtain an analogous improvement to (30) and

X

kkk=1

kΩbb^kk²₁1/2

≤Cε^3/2. (34)

In addition to these bounds, we also obtain that the map Bε⊂H^s+1×H^s→H¹: (u(·,0), ∂tu(·,0))7→bc(0)

(withBε the ball of radiusεcentered at 0) is Lipschitz continuous with a Lipschitz constant proportional toε⁻¹: att= 0,

k|ba₂−ab₁k|²₁+k|Ω(bb₂−bb₁)k|²₁≤ C ε²

ku2−u1k²_s+1+k∂tu2−∂tu1k²_s . (35)

3.8. Defects

For the functionsz_j^kobtained as the 4N-th iterate of the reverse Picard iteration of Section 3.3, we consider the defectd= (d^k_j) in (15),

d^k_j = ω_j²−(k·ω)²

z^k_j + 2iε(k·ω) ˙z_j^k + ε²z¨_j^k (36) +Fj

XN m=2

1

m!g^(m)(0) X

k1+···+km=k

z^k1. . . z^km. This is to be considered forkkk ≤N K, where we setz_j^k= 0 forkkk> K= 2N. The approximationeugiven by (2) inserted into the wave equation (1) yields the defect

δ=∂_t²eu−∂²_xue+ρeu+g(u)e (37) with

δ(x, t) = X

kkk≤N K

d^k(x, εt) e^i(k·ω)t+RN+1(eu(x, t)), (38) where RN+1 is the remainder term of the Taylor expansion of g after N terms. By (32), we havekRN+1(u)ke s+1≤Cε^N+1. We need to bound

X

kkk≤N K

d^k(·, εt) e^i(k·ω)t

2 s=

X∞ j=−∞

ω_j^2s X

kkk≤N K

d^k_j(εt) e^i(k·ω)t

2

≤CN K,1

X∞ j=−∞

X

kkk≤N K

ω^2s_j ω^|k|d^k_j(εt)

2

(39)

=CN K,1

X

kkk≤N K

ω^|k|d^k(·, εt)

2 s.

For the inequality we have used (17) with 1 in place ofs and the Cauchy- Schwarz inequality. In the next three subsections we estimate the right- hand side of (39) by Cε^2(N+1), separately for truncated modes kkk > K and near-resonant modes (j,k)∈ Rε, where z^k_j = 0 in both cases, and for non-resonant modes withz^k_j constructed above.

3.9. Defect in the truncated modes Forkkk> K we havez_j^k= 0, and the defect reads

d^k_j =Fj

XN m=2

g^(m)(0) m!

X

k¹+···+km=k

z^k1. . . z^km=ε^[[k]]ω^−|k|f_j^k

withk|fk|²_s≤Csεby (29) and (21), used withN K in place ofK. We then

have X

kkk>K

X∞ j=−∞

ω_j^2sω^|k|d^k_j²= X

kkk>K

X∞ j=−∞

ω^2s_j f_j^k2ε^2[[k]]

and hence, since 2[[k]] =kkk+ 1≥K+ 2 = 2(N+ 1), X

kkk>K

X∞ j=−∞

ω_j^2sω^|k|d^k_j²≤Csε^2(N+1). (40)

3.10. Defect in the near-resonant modes

For (j,k) in the set Rε of near-resonances defined by (4) we have set z_j^k= 0. The defect corresponding to the near-resonant modes is thus

d^k_j =Fj

XN m=2

g^(m)(0) m!

X

k1+···+km=k

z^k1. . . z^km=ε^[[k]]ω^−s|k|fb_j^k withk|bfk|²₁≤C1εby (33) and (24). We then have

X

(j,k)∈Rε

ω_j^2sω^|k|d^k_j²= X

(j,k)∈Rε

ω_j^2(s−1)

ω^2(s−1)|k|ε^2[[k]]ω_j²|fb_j^k|²

≤C1 sup

(j,k)∈Rε

ω_j^2(s−1)ε^2[[k]]+1 ω^2(s−1)|k| .

Condition (5) is formulated such that the supremum is bounded byC₀²ε^2(N+1),

and hence X

(j,k)∈Rε

ω_j^2sω^|k|d^k_j²≤Cε^2(N+1). (41)

3.11. Defect in the non-resonant modes

The scaled defect (36), as it appears in (39), reads as follows in terms of c=a+b=a^(4N)+b^(4N)defined in the iteration (26), which corresponds to the rescaling (20):

ω^|k|d^k_j = ω_j²−(k·ω)²

ε^[[k]]c^k_j+ 2i(k·ω)ε^1+[[k]]c˙^k_j + ε^2+[[k]]¨c^k_j+ε^[[k]]f_j^k(c).

(42) Expressing, for the casesk=±hjiandω_j− |k·ω|> ε^1/2, the nonlinearity in terms of the functionsFandGof the iteration (26), we find

ωjd^±hji_j =±2iωjε²

˙

a^±hji_j (4N)

−

˙

a^±hji_j (4N+1)

(43) ω^|k|d^k_j = ω²_j −(k·ω)²

ε^[[k]]

b^k_j(4N)

−

b^k_j(4N+1)

(44) with the second formula again valid forω_j− |k·ω|> ε^1/2. This suggests to reconsider the iteration (26) in the transformed variablesea andbe given as

e

a^±hji_j = (αa)^±hji_j :=±iε²a^±hji_j eb^k_j = (βb)^k_j := ω_j²−(k·ω)²

ε^[[k]]b^k_j.