On the energy distribution in Fermi Pasta Ulam lattices

Transcription

1 Version of 3 January 01 On the energy distribution in Fermi Pasta Uam attices Ernst Hairer 1, Christian Lubich 1 Section de mathématiques, -4 rue du Lièvre, Université de Genève, CH-111 Genève 4, Switzerand. E-mai: Ernst.Hairer@unige.ch Mathematisches Institut, Universität Tübingen, Auf der Morgenstee, D-7076 Tübingen, Germany. E-mai: Lubich@na.uni-tuebingen.de Abstract: For FPU chains with arge partice numbers, the formation of a packet of modes with geometricay decaying harmonic energies from an initiay excited singe ow-frequency mode and the metastabiity of this packet over onger time scaes are rigorousy studied in this paper. The anaysis uses moduated Fourier expansions in time of soutions to the FPU system and expoits the existence of amost-invariant energies in the moduation system. The resuts and techniques appy to the FPU α- and β-modes as we as to higher-order noninearities. They are vaid in the regime of scaing between partice number and tota energy in which the FPU system can be viewed as a perturbation to a inear system, considered over time scaes that go far beyond standard perturbation theory. Weak non-resonance estimates for the amost-resonant frequencies determine the time scaes that can be covered by this anaysis. 1. Introduction This report is intended to be the first one in a series deaing with the behavior of certain noninear physica systems where the non-inearity is introduced as a perturbation to a primariy inear probem. The behavior of the systems is to be studied for times which are ong compared to the characteristic periods of the corresponding inear probem. E. Fermi, J. Pasta, S. Uam 1955 The numerica experiment by Fermi, Pasta and Uam [11], which showed unexpected recurrent behaviour instead of reaxation to equipartition of energy in a chain of weaky nonineary couped partices, has incited a weath of research in both mathematics and physics and continues to do so; see the recent voume edited by Gaavotti [14] and the review by Berman and Izraiev [7] as we as the accounts on the earier FPU history by Ford [13] and Weissert [0]. The present paper contributes to the vast iterature on FPU with an anaysis, for arge partice numbers, of the questions of the formation of a packet of modes with geometricay decaying energies, starting from a singe excited mode, and of the metastabiity shown in the perseverance of the packet over onger time scaes see Benettin, Carati, Gagani and Giorgii [3] for a review of the FPU probem from the metastabiity perspective. Ony recenty, such questions have been addressed anayticay in an impressive paper by Bambusi and Ponno [1]. Here we present an approach to these questions that differs substantiay: in the scaing between partice number N and tota energy E considered, in the

2 E. Hairer and Ch. Lubich time scaes of metastabiity obtained, and in the kind of anaysis empoyed. The scaing considered here is such that the noninearity can indeed be viewed as a perturbation to the inear probem cf. the citation above, which is the case for E N 3 in the FPU α-mode cubic potentia, whereas in [1] the scaing E N 3 is studied. In the FPU β-mode quartic potentia we require E N 1. We derive our resuts using a non-resonant moduated Fourier expansion in time of the soution to the FPU system, as opposed to the use of an integrabe resonant norma form in [1]. The resuts and techniques deveoped here appy to FPU α- and β-modes as we as to FPU systems with higher-order noninearities, whereas those of [1] are so far restricted to the α-mode. Integrabiity is an essentia concept in [1], but pays no roe here. The moduated Fourier expansion empoyed here is a mutiscae expansion whose coefficient functions are constructed from a moduation system that retains a Hamitonian structure. The moduation system is shown to have amostinvariants that majorize the norma mode energies of the FPU system. This permits us to expain the preservation of the ow-frequency packet over time scaes that are much onger than the time scae for the formation of the packet. We mention that moduated Fourier expansions have been used simiary before in the anaysis of numerica methods for osciatory Hamitonian differentia equations [17, 8] and in the ong-time anaysis of non-ineary perturbed wave equations [9] and Schrödinger equations [15, 16]. For previous numerica experiments that are in reation to the present work, we refer to De Luca, Lichtenberg and Ruffo [10], Berchiaa, Gagani and Giorgii [6], and Fach, Ivanchenko and Kanakov [1]. The paper is organized as foows: In Section we introduce the necessary notation and formuate the probem. Section 3 presents numerica experiments that iustrate the main resut, Theorem 1, which is stated in Section 4. This theorem provides rigorous bounds on the geometric decay of the energies of a modes and on the ong-time preservation of the packet, in the case of the FPU α-mode with sufficienty sma initia excitation in the first mode. The proof of Theorem 1 is given in Sections 5 to 8. Section 5 provides the necessary weak non-resonance estimates for the frequencies of the FPU system. These are needed for the construction of the non-resonant moduated Fourier expansion given in Section 6. Some further bounds for the moduation functions are derived in Section 7. In Section 8 we construct the amost-invariant energies of the moduation system and show that they bound the energies in the modes of the FPU system. We are then abe to bound the mode energies over onger time scaes than the vaidity of the moduated Fourier expansion and compete the proof of Theorem 1. We obtain even onger time scaes by incuding certain high-order resonances among frequencies in the construction of the moduated Fourier expansion, which is done for the first appearing resonance in Section 9. Finay, the extension of Theorem 1 to the FPU β-mode and higher-order modes is given in Section 10.. Formuation of the probem The periodic Fermi Pasta Uam attice with N partices has the Hamitonian H = N n=1 1 p n + N 1 n=1 q n+1 q n + V q n+1 q n,

3 Long-time energy distribution in FPU attices 3 where the rea sequences p n and q n are N-periodic, and has the equations of motion q n q n+1 q n + q n 1 = V q n+1 q n V q n q n 1. 1 The non-quadratic potentia is typicay taken as V x = αx 3 /3 + βx 4 /4. With the discrete Fourier coefficients 1 u = u j, q n = u j e ijnπ/n, we obtain, for the specia case α = 1 and β = 0, the system ü j + ωj u j = i ω j 1 j1+j j/n ω j1 ω j u j1 u j 3 with frequencies j 1+j =j mod N jπ ω j = sin, j = N,..., N 1. 4 N Equation 3 is a compex Hamitonian system of the form ü j + ω j u j = j Uu 5 j denotes derivative with respect to u j with potentia Uu = i 3 j 1+j +j 3=0 mod N The tota energy of the FPU system is E = 1 j1+j+j3/n ω j1 ω j ω j3 u j1 u j u j3. 6 where E j is the jth norma mode energy, E j = N ɛ j with ɛ j = 1 uj 1 + E j + N U, 7 ω j uj. 8 Our interest in this paper is in the time evoution of these mode energies, and we write E j t = E j ut, ut. Since the q n are rea, we have u j = u j, and hence E j = E j for a j. Occasionay it wi be convenient to work with the specific energy ɛ = E N. 9 We are mainy interested in sma initia data that are different from zero ony in a singe pair of modes ±j 0 0: u j 0 = u j 0 = 0 for j ±j In contrast to much of the FPU iterature we omit the normaization factor 1/ N in. With this scaing there is no factor N in the system 3 and no factor N in the potentia 6, but the factor N appears in the energies 7 and 8.

4 4 E. Hairer and Ch. Lubich 10 0 N = 8 E = N 3 0 t 10 N N = 3 E = N 3 0 t 10 N N = 18 E = N 3 0 t 10 N Fig. 1. Norma mode energies E j t as functions of time for the FPU α-probem with initia data of Section 3; increasing j corresponds to decreasing vaues of E j t. Equivaenty, ony the j 0 th mode energy is non-zero initiay. Because of ü 0 = 0 which foows from ω 0 = 0 this aso impies that u 0 t = 0 for a t. This property is vaid for genera potentias V, because summing up the equation 1 over a n shows that the second derivative of n q n vanishes identicay. Remark 1. The origina artice [11] considers the differentia equation 1 with fixed boundaries q N = q 0 = 0. Extending such data by q j = q j to a Nperiodic sequence, we notice that the extension is sti a soution of 1, so that a statements of the present artice remain vaid aso in this situation. 3. Numerica experiment For our numerica experiment we consider the potentia V x = x 3 /3, i.e., α = 1 and β = 0. For N an integra power of, we consider initia vaues

5 Long-time energy distribution in FPU attices N = 8 E = N 5 0 t 10 N N = 3 E = N 5 0 t 10 N Fig.. Norma mode energies E j t as functions of time for the FPU α-probem with initia data as in Fig. 1, but with smaer tota energy E = N 5. q n 0 = ɛ ω 1 sin x n, q n 0 = ɛ cos x n 11 with x n = nπ/n, so that 10 is satisfied with j 0 = 1 and the tota energy is E = Nɛ. The soution is computed numericay by a trigonometric method treating without error the inear part of the differentia equation with step size h = 0.01 for N = 8, with h = 0.1 for N = 3, and with h = 1 for N = 18. Figure 1 shows the mode energies E j t as functions of time for E = N 3. This is the situation treated in [1]. Figure shows the mode energies for E = N 5. In the figures we have chosen time intervas that are proportiona to N 3, which ooks ike a natura time scae for the sow changes in the mode energies. Further numerica experiments with many different vaues of E and N indicate that, with sma the mode energies behave ike δ = E N 3 1, 1 E j t E δ j 1 f j N 3 t for j 13 with functions f j τ that do not depend significanty on E and N and which are of size c j 1 with c < 1. We have E 1 0 = E 1 0 = E/ and we observe that E 1 t E 1 0 E δ f N 3 t.

6 6 E. Hairer and Ch. Lubich 4. Main resut The proof of our main resut on the ong-time behaviour of norma mode energies is confronted with sma denominators, and reies on ower bounds of the form ω j ω j r + N =1 k ω γ π3 ω j N 3, 14 where r and N =1 k are sma integers, but j can be arbitrariy arge. Because of the specia form 4 of the frequencies ω j, such an estimate can be obtained for neary a reevant situations, as wi be eaborated in Section 5 beow. The ony difficuty arises when there exists an integer s such that the expression j rπ cos s 15 4N r is very sma. We therefore restrict the admissibe dimensions N to those beonging to the non-resonance set N M, γ of the foowing definition, where the integer M wi determine the time scae t cn δ M 1 with δ 1 of 1 over which we obtain a geometric decay of the mode energies, whie γ > 0 is from the non-resonance condition 14. Definition 1. Let an integer M and a constant γ > 0 be given. The dimension N beongs to the non-resonance set N M, γ if for a pairs of integers s, r with 0 < s < r M and even r s, for the integer j minimizing 15, and for a k 1,..., k N satisfying N =1 k M and N =1 k = s, condition 14 hods true. The foowing statements on the non-resonance set N M, γ are obtained by numerica investigation: M = : A dimensions N beong to N, γ for every γ, because for the ony candidate s, r = 1, the difference r s is not an even number. M = 3: We have to consider s, r = 1, 3. For γ = 0.1 the set N 3, γ contains a N between 100 and except for N = 78. This vaue of N beongs to N 3, γ for γ = M = 4: In addition to the pair 1, 3 we have to consider s, r =, 4. Since cos π 3 = 1, the integer j minimizing 15 is the integer that is cosest to N 3 + s. Whenever N is an integra mutipe of 3 we have exact resonance, and condition 14 cannot be fufied with a positive γ. A other N with the exception of N = 78 beong to N 4, γ with γ = 0.1. M = 5: For γ = 0.1 the set N 5, γ contains a vaues of N in the range 100 N , except for integra mutipes of 3 and 37 additiona vaues. We are now in the position to formuate our main resut. We assume that initiay ony the first pair of modes ±1 is excited, that is, the initia data satisfy 10 for j 0 = 1. Theorem 1. Fix γ > 0 and ρ 1 arbitrariy, and et M and K be positive integers satisfying M < K and K +M = 10. Then, there exist δ 0 > 0 and c > 0, C > 0 such that the foowing hods: if

7 Long-time energy distribution in FPU attices 7 1 the dimension of the system satisfies N 41 and N N M, γ, the tota energy E is bounded such that δ := E N 3 δ 0, 3 the initia norma mode energies satisfy E j 0 = 0 for j ±1, then, over ong times t c N δ M 1, the norma mode energies satisfy the estimates E 1 t E 1 0 C E δ, 16 E j t C E δ j 1, j = 1,..., K, 17 N ρ j E j t C E δ K j=k This theorem is proved in Sections 5 through 8. Moreover, the proof shows that the theorem remains vaid if the singe-mode excitation condition 10 is repaced by the geometric-decay condition that 17 and 18 hod at t = 0, with a given constant C 0 in pace of C and a ρ 0 > ρ in pace of ρ. We note that the maxima time interva is obtained with M = 4, for which we have t c N δ 5. Longer time intervas for sighty weaker estimates wi be obtained in Section 9, where we account for the amost-resonance ω 5 ω 4 + 3ω 1 = ON 5 that eads to the restriction K + M 10 in the above theorem. The decay of the mode energies with powers of δ was first observed numericay by Fach, Ivanchenko and Kanakov [1]. To our knowedge, so far the ony rigorous ong-time bounds of the mode energies in FPU systems with arge partice numbers N have been given by Bambusi and Ponno [1]. There it is shown for E = N 3 and arge N that the mode energies satisfy E j t EC 1 e σj + C N 1 for t T N 3, where the factor T can be fixed arbitrariy, and the constants C 1, C and σ depend on T. In comparison, our resut is concerned with the case E N 3. In this situation we obtain exponentia decay for a j and estimates over much onger time intervas. For E = δ 0N 3, however, our estimates are proved ony on time intervas of ength ON. The proof of [1] expoits the reationship of the FPU α-mode with the KdV equation for the scaing E = N 3. The proof of Theorem 1 is competey different and is based on the technique of moduated Fourier expansions. The proof of Theorem 1 gives expicit formuas for the dominant terms in E j t: E j t = E δ j 1 c j + c + j a j t + Oδ + N, 19

8 8 E. Hairer and Ch. Lubich where c ± = 1 ɛ ω1 u 1 0 ± i u 1 0 with ɛ = E/N, so that c + c + = 1, and the first functions a j t are given by a 1 t = 1 a t = 1 e iω1 ωt π 1 a 3 t = 1 3 π 4 e i3ω1 ω3t 1 e iω1+ω ω3t 1. A the inear combinations of frequencies appearing in these formuas are of size ON 3. In particuar, we have ω 1 ω = π3 4 N 3 + ON 5. The function a t is thus periodic with period T 8π N 3. This agrees extremey we with Figure, where the choice 11 of the initia vaues corresponds to c + = 0 and c = i. We remark that the mode energy vaues E j E j δ/π j 1 stated in [1] are quaitativey simiar, though not identica in quantity to the expressions given here. 5. Weak non-resonance inequaities The frequencies ω j = sin jπ N are amost in resonance for arge N: jπ π j + π ω j + ω ω j+ = 8 sin sin sin = ON 3 4N 4N 4N for sma j and. Such a near-resonant situation eads to sma denominators in the construction of the moduated Fourier expansion that is given in the next section. We therefore present a series of technica emmas that dea with the amost-resonances among the FPU frequencies. We consider muti-indices k = k 1,..., k N with integers k, we define µk = N k, =1 and we denote j = 0,..., 0, 1, 0,..., 0 the j -th unit vector. Furthermore, we write ω = ω 1,..., ω N and k ω = k 1 ω k N ω N. Our approach with moduated Fourier expansions eads to sma denominators ωj k ω which have to be bounded from beow. We first consider j, k with sma j and sma µk. Lemma 1. For pairs j, k satisfying max j, µk 10 and k ± j, we have for N 7 π N ω j k ω ω j + k ω if =1 4 N k ±j π 3 8 N 3 ω j + k ω ese.

9 Long-time energy distribution in FPU attices 9 The above estimate aso hods for µk = 11 except for j, k = ±5, ±3 1 4, for which ω 5 3ω 1 ω 4 = ON 6. Proof. For arge N we expand the expression ω j k ω into a Tayor series, use the remainder bounds sin x x x 3 /6 and sin x x+x 3 /6 x 5 /10, and thus obtain the estimates for N 100. We empoy the fact that for µk 10 the reation j + k = 0 impies j k 6 checked numericay. Aso the remaining finitey many cases are checked numericay. We next consider the expression ω j ω + k ω, where is arge and j and µk are sma. Lemma. For pairs j, + k satisfying j 11, 7, µk 5, and + k ± j, we have for N 41 π ω j ω + k ω ω j + ω + k ω if + N =1 8 N k ±j π 3 8 N 3 ω j + ω + k ω ese. Proof. For arge 0 = 0, for 0 j 11, µk 5, and N 37, we prove ω + k ω ω j ω 0 + k ω ω j π N. The second inequaity is obtained from 0 + N =1 k j 4 > 1 by estimating the remainder in the Tayor series expansion. The finitey many remaining cases are verified numericay. We finay consider the expression ω j ω j r +k ω for arbitrariy arge j, but with sma r and sma µk. The factor ω j +ω j r +k ω wi be bounded from beow by ω j and the expression ω j ω j r + k ω is given by r π j rπ 4 sin cos π 4N 4N N N k + =1 π3 4 N 3 N =1 1 3 k + O N 5. 0 Lemma 3. Let M, r be integers such that M 15 and r M, and consider pairs j, j r + k satisfying M < j minn, N + r, µk M, and j r + k ± j. With s = k we then have for a dimensions satisfying N max π 7/ M, π M 3 + 6/1 that π ω j if s < minr, 0 or s > maxr, 0 ω j ω j r + k ω N π 3 ω j 8 N 3 if s = r = 0 r π ω j 8 N if s = 0 and r 0 r π ω 3 j 3 N if s = r and r 0. 1 A counter-exampe for the second estimate is k ω = ω 5 ω 4 + 3ω 1 = ON 5, for which µk = 16.

10 10 E. Hairer and Ch. Lubich Proof. The assumptions on the indices impy j r N r. The first term in 0 is thus a monotonic function of j with asymptotic vaues ranging from r π N r r π 4 N when j goes from j = M to j = maxn, N + r. This observation impies the first inequaity of 1, because the second term in 0 is dominant in this case. Rigourous estimates prove the inequaity for N π M 3 /6. The second inequaity foows as in Lemma 1 for N π M 5 /3840 from the fact that for 0 < µk 15 the condition k = 0 impies 3 k 6. For s = 0 and r 0, the first term in 0 is dominant and therefore we have the third inequaity of 1 for N πm 3 + 6/1. For the proof of the ast inequaity we note that jπ r π j rπ ω j ω j r ω r = 8 sin sin sin 4N ω j r π sin 4N 4N ω j cos r π 4N 4N 4 sin r π 4N where we have used the addition theorem for sinα β and the inequaity sinα/ sin α. We therefore obtain with χ j = ω j 4 ω j ω j r k ω r π ω j 3 N + χ j sin r π N r π ω j r π 8 N 8 N ω r k ω. To prove χ j 0, we notice that it is an increasing function of j for j M, and bounded from beow by its vaue at j = M. For r = k and µk M we have k ± k M ± r, so that k = 0 for > M + r, and therefore 3 k MM + r /4. Using x x 3 /6 sin x x we obtain the ower bound χ M r 3 π 3 M 4 N 3 A r 3 M π 5 M B r 48 N 5 r with Ax = 9 x 6 x 1 xx + 1 /4, Bx = x. An inspection of these functions shows that the quotient satisfies 0 < Bx/Ax B1/A1 = 7 on an interva incuding 1 x 15. Hence, χ M 0 for N π M 7/, which competes the proof of the emma. It remains to consider the situation where r 0 and 0 < s/r < 1. We have a near-canceation of the first two terms in 0 if the index j minimizes 15. We denote such an integer by j s, r, N. Lemma 4. Let µk M, M < j minn, N + r, 0 < r M and 0 < s/r < 1, where s = k. For N 0.64 M 3 and j j s, r, N, we have the ower bound ω j ω j r + k ω π ω j 8 N r.

11 Long-time energy distribution in FPU attices 11 α+β α β, and we use cos β cos α = sin sin and from Proof. We define α by cos α = s r with β = j rπ π 4N. The estimate then foows from α β 4N sin α+β sin α = 1 cos α = r s r r. The most critica case for a ower bound of ω j ω j r + k ω is when j = j s, r, N. This is why we restrict the dimension N of the FPU-system to vaues satisfying the non-resonance condition of Definition 1. Because of property 4 beow we consider ony even vaues of r s in Definition Moduated Fourier expansion Our principa too for studying the ong-time behaviour of mode energies is a moduated Fourier expansion, which was originay introduced for the study of numerica energy conservation in Hamitonian ordinary differentia equations in the presence of high osciations [17, 8]. This technique was aso successfuy appied to the ong-time anaysis of weaky noninear Hamitonian partia differentia equations [9, 15, 16]. The idea is to separate rapid osciations from sow variations by a two-scae ansatz of the form ω j u j t zj k τ e ik ωt with τ = N 3 t, k K j where K j is a finite set of muti-indices k = k 1,..., k N with integers k. The products of compex exponentias e ik ωt = N =1 eik ω t account for the noninear interaction of different modes. The sowy varying moduation functions zj k τ are yet to be determined from a system of moduation equations Choice of the interaction set. We wi choose the sets K j such that the interaction of ow modes with ow modes and high modes with ow modes is incorporated, but the interaction of high modes with high modes is discarded. It turns out that for our purposes an appropriate choice of the muti-index set K j is obtained as foows: fix positive integers K and M with K + M 10 and M < K. We define, with the notation introduced at the beginning of Section 5, K = {j, k : max j, µk K + M} {j, ± + k : j K + M, µk M, K + 1} {j, ± j r + k : j K + M, r M, µk M}. This set consists of pairs j, k for which we have obtained ower bounds for ω j k ω in Section 5. We et K j = {k : j, k K}. For convenience we set z k j τ = 0 for muti-indices k that are not in K j.

12 1 E. Hairer and Ch. Lubich 6.. Choice of norm. We work with a weighted norm for N-periodic sequences u = u j, u = σ j u j, σ j = Nω j s ρ Nωj 3 with s > 1/ and ρ 1. This choice is motivated by two facts. On the one hand, the extended norm u, u = Ωu + u 4 with Ω = diag ω j can be written in terms of the mode energies as ut, ut = 1 N N σ j E j t, where the prime indicates that the ast term in the sum is taken with the factor 1/. On the other hand, with the given choice of σ j, the norm behaves we with convoutions. Lemma 5. For the norm 3, we have j=1 u v c u v where u v j = with c depending on s > 1/, but not on N and ρ 1. j 1+j =j mod N u j1 v j, 5 Proof. We note the bound ω j ω j1 + ω j for j 1 + j = j mod N, which foows from the addition theorem for sinα + β. Together with the inequaity j N ω j π j, this yieds σ 1 σ 1 c σ 1 j. j 1+j =j mod N The resut then foows with the Cauchy-Schwarz inequaity: σ j u v j σ j σ 1 j 1 σ 1 j σ j1 u j1 σ j v j, j j j 1+j j j 1+j j j 1 j where means congruence moduo N Statement of resut. The approximabiity of the soution of the FPU system by moduated Fourier expansions is described in the foowing theorem. Theorem. Fix γ > 0, ρ 1, s > 1/, and et M and K be positive integers satisfying M < K and K + M = 10. Then, there exist δ 0 > 0 and C 0 > 0, C 1 > 0, C > 0 such that the foowing hods: if 1 the dimension of the system satisfies N 41 and N N M, γ, the energy is bounded such that δ := E N 3 δ 0,

13 Long-time energy distribution in FPU attices 13 3 the initia vaues satisfy E 0 0 = 0 and K j N E j 0 C 0 E δ j 1 for 0 < j < K, K 1 σ j E j 0 C 0 E δ 6 with the weights σ j = Nω j s ρ Nωj from 3; then, the soution ut = u j t ω j u j t = where the remainder rt = r j t admits an expansion k K j z k j τ e ik ωt + ω j r j t with τ = N 3 t, 7 is bounded in the norm 4 by rt, ṙt C ɛ δ K+M N t for 0 t minn 3, ɛ 1/ 8 with the specific energy ɛ = E/N. The moduation functions zj k τ are poynomias, identicay zero for j = 0, and for 0 τ 1 bounded by K j N zj k τ C1 ɛ δ j 1 for 0 < j < K, k K j σ j zj k τ C1 ɛ δ K 1. k K j 9 The same bounds hod for a derivatives of zj k with respect to the sow time τ = N 3 t. Moreover, the moduation functions satisfy z k j = zk j. The above estimates impy, in particuar, a soution bound in the weighted energy norm: for t minn 3, ɛ 1/, E j t C 1 E δ j 1 for 0 < j < K, σ j E j t C 1 E δ K 1. K j N 30 We sha see ater that such estimates actuay hod on much onger time intervas. The rest of this section is devoted to the proof of Theorem Forma moduation equations. Formay inserting the ansatz into 3 and equating terms with the same exponentia e ik ωt ead to a reation ω j k ω z k j + i k ω N 3 dzk j dτ + N 6 d z k j dτ =..., 31 where the dots, coming from the non-inearity, represent terms that wi be considered ater. Since we aim at constructing functions without high osciations, we have to ook at the dominating terms in 31.

14 14 E. Hairer and Ch. Lubich For k = ± j, where j = 0,..., 0, 1, 0,..., 0 with the non-zero entry in the j -th position, the first term on the eft-hand side of 31 vanishes and the second term with the time derivative dzj k /dτ can be viewed as the dominant term. Reca that we ony have to consider j 0. For k K j and k ± j, the first term is dominant according to the non-resonance estimates of Section 5. To derive the equations defining the coefficient functions zj k, we have to study the non-inearity when is inserted into 3. We get for k = ± j that dz± j 3 j ±i ω j N + N 6 d z ± j j dτ dτ 3 and for k ± j = i ω j j 1+j =j mod N 1 j1+j j/n k 1 +k =± j zj k1 1 zj k, ω j k ω zj k + i k ωn 3 dzk j dτ + N 6 d zj k dτ 33 = i ωj 1 j1+j j/n zj k1 1 zj k. j 1+j =j mod N k 1 +k =k In addition, the initia conditions for u j need to be taken care of. They wi yied the initia conditions for the functions z ± j j for j 0: k K j z k j 0 = ω j u j 0, k K j i k ω z k j 0 + N 3 dzk j dτ 0 = ω j u j Construction of coefficient functions for the moduated Fourier expansion. We construct the functions zj k τ such that they are soutions of the system 3 33 up to a sma defect and satisfy the initia conditions 34 for a j. We write the functions as a forma series in δ = EN 3, z k j τ = ɛ m 1 δ m 1 z k j,mτ = N m 1 δ m z k j,mτ, 35 insert them into the reations 3 34, and compare ike powers of δ: dz± j 3 j,m ±i ω j N + N 6 d z ± j j,m dτ dτ 36 = i ωj N 1 j1+j j/n and for k ± j, j 1+j =j mod N ω j k ω zj,m k + i k ωn 3 dzk j,m dτ = i ωj N zj k1 1,m 1 zj k,m, k 1 +k =± j m 1+m =m + N 6 d zj,m k dτ 37 1 j1+j j/n zj k1 1,m 1 zj k,m. j 1+j =j mod N k 1 +k =k m 1+m =m

15 Long-time energy distribution in FPU attices 15 This yieds an equation for zj,m k k ± j and for the derivative of z± j j,m with a right-hand side that depends ony on the product zj k1 1,m 1 zj k,m with m 1 +m = m so that both m 1 and m are stricty smaer than m. Initia vaues for z ± j j,m are obtained from 34. These differentia equations possess a unique poynomia soution, since the ony poynomia soution to with a poynomia p of degree d is given by z = z α dz dτ β d z dτ = p 38 d α d p. dτ + β d 39 dτ =0 We now show in detai how the coefficient functions zj,m k τ for the first few vaues of m = 1,,... are constructed. According to the bound 6 we assume for 1 j K that ω j u j 0 = ɛ δ j 1 a j, u j 0 = ɛ δ j 1 b j 40 consequenty ω j u j 0 = ɛ δ j 1 a j, u j 0 = ɛ δ j 1 b j with factors a j and b j whose absoute vaues are bounded independenty of N, ɛ, and δ. Case m = 1: For m = 1, the right-hand sides in 36 and 37 are zero. The reation 37 shows that zj,1 k τ = 0 for k ± j. Equation 36 shows that d dτ z± j j,1 τ = 0 and hence z± j j,1 τ is a constant function. Its vaue is obtained from 34. It vanishes for j ±1, and is given by z ± 1 1,1 0 = 1 a 1 ib 1, z ± 1 1,1 0 = 1 a 1 ib 1 for the ony non-vanishing functions with m = 1. Case m = : The products z ± 1 1,1 z± 1 1,1 and z ± 1 1,1 z± 1 1,1 with a combinations of signs give a non-zero contribution to the right-hand side of 37. Notice that j = 0 need not be considered, because u 0 = 0. This gives the constant functions z 0,τ = i z 1 1,1 0z 1 1,1 0 N ω 4ω1 1 z, τ = i ω z 1 1,1 0z 1 1,1 0 N and simiar formuas for z,τ, 0 z 1, τ, and z ± 1, τ. From 36 we obtain d dτ z±, τ = 0, and hence z±, τ is constant. The initia vaues are determined from 34, i.e., z, 0 + z, 0 + z 1, 0 + z 1, 0 = a i z, 0 z, 0 + i ω 1 1 z, ω 0 z 1, 0 = b, since the vaues z ± 1, 0 are aready known. In the same way we get z ± These functions are the ony non-vanishing functions for m =., τ.

16 16 E. Hairer and Ch. Lubich Case m = 3: The nonzero terms in the right-hand side of 37 are formed of products zj k1 1,m 1 zj k,m, where one index among m 1, m is 1 and the other is. We thus obtain formuas for z ±3 1 1,3 τ, z ± 1 ± 1,3 τ, z ±3 1 3,3 τ, z ± 1 ± 3,3 τ, τ, and simiar formuas with negative index j. A these functions are z ± 1 3,3 constant. The reation 36 eads to differentia equations with vanishing righthand side for z ± 3 3,3 τ and with constant right-hand side for z± 1 1,3, which thus becomes a inear poynomia in τ. Case 4 m < K: Assuming that the functions zj k1 1,p are known for p m 1 and a j 1 and k 1, the reation 37 permits us to compute zj,m k for k ± j. We further obtain z ± j from 36 and 34. By this construction many of the j,m functions are constant and some of them are poynomias in τ, of degree at most m/. Case K m K + M with M < K: A new situtation arises for m = K, because for j K we have, according to the bound 6, that ω j u j 0 = ɛ δ K 1 a j, u j 0 = ɛ δ K 1 b j 41 with a j, b j = O1. Hence, for m = K, we get differentia equations for a diagona functions z ± j j,m with initia vaues that in genera are not zero. Otherwise, the construction can be continued as before. Since a coefficient functions are created from diagona quantities z j j τ, we have for a m 1 that Furthermore, as ong as m K, we have z k j,mτ = 0 z k j,mτ = 0 if j µk mod. 4 if { m < j or m < µk or m j mod or m µk mod Bounds of the moduation functions. The above construction yieds functions zj,m k that are bounded by certain non-positive powers of N. From the expicit formuas of Section 6.5 we have z ± 1 1,1 τ = O1 z ±, τ = O1, z± 1, τ = O1, z,τ 0 = ON z ± 3 3,3 τ = O1, z±3 1 3,3 τ = O1 z ± 1 ± 3,3 τ = O1, z ± 1 3,3 τ = ON z ± 1 1,3 τ = ON + Oτ, z ± 1 3,3 τ = ON z ±3 1 1,3 τ = ON, z ± 1 ± 1,3 τ = ON, z ± 1 1,3 τ = O1

17 Long-time energy distribution in FPU attices 17 and simiary for negative index j. In the course of this subsection we show that for m K + M, σ j zj,mτ k C for 0 τ 1 k K j with a constant C that depends on K and M, but is independent of N. j, k, m j 1, k 1, m 1 j, k, m Fig. 3. Binary decomposition. The behaviour of zj,m k can best be understood by using rooted binary trees. For muti-indices k ± j we see from 33 that zj,m k is determined by the products zj k1 1,m 1 zj k,m, where j = j 1 + j, k = k 1 + k, and m = m 1 + m, as iustrated in Figure 3. Recursivey appying this reduction, we see that zj,m k can be bounded in terms of products of diagona terms z ±,p with p < m. For exampe, z 1 3,5 contains a term z 1 z 1 1,1 1,3. This is iustrated by the binary tree of Figure 4. 3, 1, 5, 1, 1, 1, 3 1, 1, 1 1, 1, 1 Fig. 4. Exampe of a binary tree with root 3, 1, 5. In such rooted binary trees, there are two types of vertices: eaves of the form, ±, p and inner vertices j, k, m with k K j and k ± j which have two branches. For k K j and m K + M, we denote by Bj,m k the set of a rooted binary trees of this kind with root j, k, m. We note that the number of trees in Bj,m k is independent of N. We estimate the moduation functions in a time-dependent norm on the space of poynomias of degree not exceeding K + M, z τ = i 0 1 i! di z. dτ i τ 43

18 18 E. Hairer and Ch. Lubich By the weak non-resonance estimates of Section 5, we have for a j, k K with k ± j that N 6 αj k 3 k ωn = i ωj k, ω βk j = ωj k satisfy α k ω j + βj k C 44 with a constant C that does not depend on j, k K and N. Since for a poynomia pτ of degree n, we have dp/dτ τ n p τ, the soution 39 of 38 is bounded by z τ Cn p τ with a constant depending on n. Equation 37 thus yieds the recursive bound, for k ± j and for m K + M, z j,m k τ γj k z j k1 1,m 1 τ z j k,m τ 45 with j 1+j =j mod N k 1 +k =k m 1+m =m γ k j = C ω j N ω j k ω, 46 where C depends on K and M, but not on N, j, and k. We resove this recurrence reation via the binary trees. We denote by κ +,p b the number of appearances of,, p and its compex conjugate,, p among the eaves of a tree b Bj,m k, and by κ,p b the number of appearances of,, p and,, p. We then obtain z k j,m τ b B k j,m Γ b N =1 m 1 p=1 z,p κ+,p b τ z,p κ,p b τ 47 with a factor Γ b that is given recursivey as foows: Γ b = 1 for a tree with a singe node j, ± j, m, and Γ b = γ k j Γ b 1 Γ b for a binary tree b having a root j, k, m with k ± j and composed of subtrees b 1 and b. Lemma 6. The estimates of Section 5 yied Γ b Const. with a constant independent of N N M, γ but depending on K, M and γ. Proof. We have γj k = O1 for a vaues of j, k with the exception of the cases corresponding to the second inequaity of 1 or to that of 14, where we ony have γj k = Oω j N which is unbounded for arge j. Nevertheess, we obtain the statement of the emma, because the non-resonance estimates of Section 5 permit to prove γ k j γ k1 j 1 γ k j = O1 for j 1 + j = j mod N and k 1 + k = k, 48 where we have set γ ± = 1 and γ k j = 0 if j K j.

19 Long-time energy distribution in FPU attices 19 Indeed, et k = j r + h with s = h, and assume that k 1 is cose to j r, so that µk and j are bounded by M K + M. We distinguish two situations. If s = k satisfies s ±j, it foows from the first inequaity of Lemma 1 that γj k = ON. This is sufficient to get 48. If s = j or s = j, so that γj k = O1, an inspection of the possibe vaues for j 1, k 1 shows that γj k1 1 = ON 1. Aso in this case we have 48. By 45 and 47 the functions zj,m k τ with k ± j are estimated in terms of the diagona functions z ±,p τ with p < m. With simiar arguments we find that the derivative of z ± j j,m τ, given by 36, is bounded by d dτ z± j j,m τ ω jn j 1+j =j mod N k 1 +k =± j m 1+m =m z j k1 1,m 1 τ z j k,m τ. 49 Here we note that the factor ω j N, which is arge for arge j, is compensated by the factors present in zj k1 1,m 1 and zj k,m. This is a consequence of γ k1 j 1 γ k j = Oω j N 1 for j 1 + j = j mod N and k 1 + k = ± j, 50 which foows from the non-resonance estimates of Section 5 exacty as in the proof of Lemma 6. The estimate z τ z0 + dz dτ τ for τ 1 together with an induction argument thus yieds z ± j j,m τ z± j j,m 0 + C where B ± j j,m same meaning as in 47. b B ± j j,m N =1 m 1 p=1 z,p κ+,p b τ z,p κ,p b τ, 51 is the set of binary trees with root j, ± j, m, and κ±,p b have the We arrive at the point where we have to estimate the initia vaue z ± j j,m 0, defined by 34. It is bounded in terms of the scaed quantities a j, b j, zj,m k 0 d for k ± j, and of boundedness of z ± j j,m and m = K the function z ± j j,m dτ zk j,m 0 for a k K j. Using 49 this impies the 0 for j K +M and m K +M. For arge j > K +M z ± j j,m 0 = 1 a j ib j, τ is constant, and we have z ± j j,m 0 = 1 a j ib j, because in this case zj,m k 0 = 0 for a k ± j. By induction we thus obtain for j > K + M and K m K + M that z ± j j,m 0 C j M a + b. By 51 we obtain a simiar estimate for z ± j j,m τ, and by 47 for a zk j,m τ. Using σ j a j + b j C,

20 0 E. Hairer and Ch. Lubich which foows from the assumption 6 on the initia vaues, we obtain the foowing resut. Lemma 7. For m K + M, we have for τ 1 σ j Together with the observation that k K j z k j,m τ C. 5 z k j,m = 0 for m < min j, K, 53 Lemma 7 yieds 9. Anaogous bounds are obtained for the derivatives. As a further consequence of this reduction process by binary trees, we note that z k j,m = 0 if k 0 and m < min, K, 54 and the ony non-vanishing coefficient functions for m = min, K are the diagona terms z ±,m Defect in the moduation equations. We consider the series 35 truncated after K + M terms, z k j τ = 1 N m K+M As an approximation to the soution of 3 we thus take ω j ũ j t = δ m z k j,mτ. 55 k K j z k j τ e ik ωt. 56 The construction is such that ũ j 0 = u j 0 and ũ j 0 = u j 0 for a j. The functions zj k τ do not satisfy the moduation equations 3-33 exacty. We denote the defects, for j = N,..., N 1 and arbitrary muti-indices k = k 1,..., k N, by d k j = ωj k ω zj k + i k ωn 3 dzk j dτ + N 6 d zj k dτ 57 + i ωj 1 j1+j j/n zj k1 1 zj k. j 1+j =j mod N k 1 +k =k By construction of the coefficient functions z k j,m, the coefficients of δm in a δ- expansion of the defect vanish for m K + M. We thus have ω j d k j = j 1+j =j mod N 1 j1+j j/n k 1 +k =k K+M m=k+m+1 m 1+m =m z k1 j 1,m 1 z k j,m δ m N 4. 58

21 Long-time energy distribution in FPU attices 1 Using the estimate 5 of the moduation functions and Lemma 5, the defect is thus bounded by σ j ω j d k j τ C δ K+M+1 N 4 k With 50, a stronger bound is obtained for the diagona defects: for 0 τ σ j ω 1 j d ± j j τ C δ K+M+1 N 5 for 0 τ The defect d k j vanishes for muti-indices j, k that cannot be decomposed as j = j 1 + j and k = k 1 + k with k 1 K j1, k K j Remainder term of the moduated Fourier expansion. We compare the approximation 56 with the exact soution u j t of 3 for initia vaues satisfying 10. With the notation of Section this yieds, for j = N,..., N 1, where the defect ϑ j t is given by ü j + ω j u j + j Uu = 0 ũ j + ω j ũ j + j Uũ = ϑ j ω j ϑ j t = k d k j τ e ik ωt with τ = N 3 t, and, by 59, bounded by σ j ω 1 j ϑ j t C δ K+M+1 N 4 for t N The initia vaues satisfy ũ j 0 = u j 0 and ũ j 0 = u j 0. The noninearity Uu = bu, u is a quadratic form corresponding to the symmetric biinear form bu, v = i ω j 1 j1+j j/n ω j1 ω j u j1 v j. j 1+j =j mod N Using Lemma 5 and ω j, we thus obtain the bound bu, v Ω 1 bu, v c Ωu Ωv. We then estimate the error r j t = u j t ũ j t by standard arguments: we rewrite the second-order differentia equation as a system of first-order differentia equations ω j ũ j = ω j ṽ j ṽ j = ω j ω j ũ j + b j ũ, ũ + ϑ j t.

22 E. Hairer and Ch. Lubich We use the variation-of-constants formua and the Lipschitz bound bu, u bũ, ũ c Ωu + Ωũ Ωu Ωũ 4 c C ɛ Ωu Ωũ for Ωu C ɛ, Ωũ C ɛ. By 61 we have ϑt Cδ K+M+1 N 4 for t N 3. The Gronwa inequaity then shows that the error satisfies the bound 8 for t minn 3, ɛ 1/. This competes the proof of Theorem. 7. Bounds in terms of the diagona moduation functions The foowing resut bounds the norms 43 of the non-diagona moduation functions zj k with k ± j in terms of those of the diagona functions z± Lemma 8. Under the assumptions of Theorem, we have the bound N z k j τ C N =1 N z for j N and k = k 1,..., k N K j, with. τ + N z τ k + N ϑ k j for τ 1, 6 σ j ϑ k j Ĉ δ K+M+1 N k K j The constants C and Ĉ are independent of E, N and τ. We reca that N z ± τ = Oδ min,k. Proof. The proof uses arguments simiar to those in Section 6.6. The defect equation 57 yieds, for k j, N zj k τ γj k N zj k1 1 τ N zj k τ + γj k ω j N d k j τ j 1+j =j mod N k 1 +k =k with γj k of 46. We denote by Bk j the set of binary trees with root j, k that are obtained by omitting the abes m at a nodes j, k, m of trees in Bj,m k for a m K + M, and by κ ± b the number of appearances of, ± among the eaves of a tree b. We then obtain the bound N zj k τ Γ b N z κ+ b τ N z κ b τ + N ϑ k j 64 0 b B k j with the same factor Γ b as in 47, and with ϑ k j satisfying the estimate 63 because of 59 and the bound γj k = Oω jn. Since every tree in Bj k contains the eaf, ± at east k times, this yieds together with z N zj k τ b B k j Γ b 1 N z = z ± τ + N z τ k + N ϑ k j. 65 Since Γ b Const. and the number of trees in Bj k impies the stated resut. is independent of N, this

23 Long-time energy distribution in FPU attices 3 We aso need another bound that foows from 64. Lemma 9. Under the assumptions of Theorem, we have the bound z k j τ C N =1 δ j z τ + z τ + ϑ k j for τ 1, 66 for 0 < j N and k K j, where ϑ k j is bounded by 63. The sum in 66 is actuay ony over with j < K + M. We reca z k j = zk j. Proof. For a tree b Bj k, et, ± be a eaf such that is argest among the eaves of b. For the other eaves i, ± i we have + i i = j and therefore j i i < K + M. Since z ± i i is bounded by ON δ i, we obtain the resut from 64. The next emma bounds the derivatives of the diagona functions in terms of their function vaues. Lemma 10. Under the assumptions of Theorem, we have the bounds z j j τ + z j j τ z j j τ + z j j τ + ϑ j for τ 1, 67 for j = N,..., N 1, with The constant C is independent of E, N and τ. σ j ϑ j C δ K+M+1 N. 68 Proof. From the defect equation 57 with k = ± j we obtain N dz± j j dτ τ ω j N j 1+j =j mod N k 1 +k =± j N z k1 j 1 τ N z k j τ The ast term is bounded by 60. We now use 65 for zj k1 1 that by 50 we have Γ b 1 Γ b = Oω j N 1 for a b 1 B k1 This gives us + ω 1 j N 5 d ± j j τ. and z k j j 1 and note and b B k j. N dz± j j C 1 dτ τ j 1+j =j mod N k 1 +k =± j 1 N z C δ N z j j τ + N z j j τ + N ϑ j, k τ + N z 1 + k τ + N ϑ j

24 4 E. Hairer and Ch. Lubich where ϑ j is bounded by 68 because of 60 and 63. For the second inequaity we note that the number of terms in the sums is independent of N, and that ±j, ± j for some combination of signs must appear among the eaves, in addition to at east two further eaves that account for the presence of the factor δ. On the other hand, we have Hence we obtain z ± j j τ z ± j j τ + dz± j j dτ τ. 1 C δ z j j τ + z j j j τ z j τ + z j j τ + ϑ j, which yieds the resut if C δ Amost-invariant energies of the moduation system We now show that the system of equations determining the moduation functions has amost-invariants that bound the mode energies E j t from above. This fact wi ead to the ong-time energy bounds of Theorem 1. The construction of the amost-invariants takes up a ine of arguments from [18, Chapter XIII] and [9, Section 4] The extended potentia. We introduce the functions yt = y k j t j,k K with ω j y k j t = z k j τ e ik ωt 69 with τ = N 3 t, so that u j t ũ j t = k yk j t. By the construction of the functions zj k from the moduation equations, the functions yk j satisfy ÿ k j + ω j y k j + i ω j j 1+j =j mod N where the defects e k j t = ω 1 1 j1+j j/n ω j1 ω j k 1 +k =k yj k1 1 yj k = e k j, 70 j d k j τ eik ωt are bounded by 59. The non- of the extended inearity is recognised as the partia derivative with respect to y k j potentia Uy given by Uy = i 3 j 1+j +j 3=0 mod N 1 j1+j+j3/n ω j1 ω j ω j3 Hence, the moduation system can be rewritten as k 1 +k +k 3 =0 yj k1 1 yj k yj k ÿ k j + ω j y k j + k j Uy = ek j, 7 where k j U is the partia derivative of U with respect to y k j.

25 Long-time energy distribution in FPU attices Invariance under group actions. The extended potentia turns out to be invariant under the continuous group action that, for an arbitrary rea vector λ = λ 1,..., λ N R N and for θ R, is given as S λ θy = e ik λθ yj k Since the sum in the definition of U is over k 1 + k + k 3 = 0, we have j,k. US λ θy = Uy for θ R. 73 Differentiating this reation wih respect to θ yieds 0 = d US λ θy = dθ θ=0 ik λ yj k k j Uy. 74 In fact, the fu Lagrangian of the system 7 without the perturbations e k j, Ly, ẏ = 1 k k ẏ k j ẏk j ω j y k j yk j Uy, is invariant under the action of the one-parameter groups S λ θ. By Noether s theorem, the corresponding Lagrangian system has a set of invariants I λ y, ẏ, which we now study as amost-invariants of the perturbed system Amost-invariant energies of the moduation system. We mutipy 7 with ik λy k j and sum over j and k. Using 74, we obtain ik λ y k j ÿk j + ωj y k j yk j = ik λy k j ek j, j k where we notice that the second terms in the sum on the eft-hand side cance. The eft-hand side then equas d dt I λy, ẏ with Hence we obtain I λ y, ẏ = d dt I λy, ẏ = k k j k ik λy k j ẏk j. 75 ik λy k j ek j. 76 In the foowing we consider the amost-invariants as functions of the moduation sequence zτ = zj k τ and its derivative dz/dττ with respect to the sow time variabe τ = N 3 t, rather than of yt defined by 69 and ẏ = dy/dt. For λ = ω, a mutipe of the th unit vector, we write dz E z, = N Iω y, ẏ = N ω I y, ẏ dτ

26 6 E. Hairer and Ch. Lubich so that dz E z, = N ω dτ k ik ω j z k j ik ωz k j + N 3 dzk j dτ. 77 By the estimates of the moduation functions we have at the initia time dz E z0, dτ 0 C0 E δ 1 for = 1,..., K, N E dz σ z0, dτ 0 C0 E δ K 1, =K 78 where C 0 ony depends on the initia vaues u0, u0. From 76 we have N 3 d dτ E dz z, dτ = i N ω k k z k j ω j d k j. 79 Theorem 3. Under the conditions of Theorem we have for τ 1 d dτ E dz zτ, dτ τ ϑ, where ϑ C E δ +K+M 1 for = 1,..., K N σ ϑ C E δ K+M 1. =K 80 Proof. We insert 58 into 79 to obtain, with L = min, K, N 3 d dτ E dz z, dτ Nω k z k j,m zk1 j 1,m 1 z k j,m δ L+K+M+1 N 6, where the sum is over a indices j, j 1, j with j 1 + j = j, muti-indices k, k 1, k with k 1 + k = k and indices m, m 1, m K + M with m L note 54 and m 1 +m K +M +1. This sum contains a number of terms that is independent of N. Estimating the moduation functions by 45 yieds d dτ E dz 8 z, Nω k γ k dτ j γk1 j 1 γj k z j ki i,m i δ L+K+M+1 N 3, where the number of terms in the sum is sti independent of N. As in 48, the non-resonance estimates of Section 5 yied i=3 ω k γ k j γk1 j 1 γj k = ON 1 for j 1 +j = j mod N and k 1 +k = k. Among the terms in the product over i, two terms satisfy j i M, and a others j i K + M. For the atter, we estimate z j ki i,m i by a constant, for the two others we use the Cauchy-Schwarz inequaity together with the estimate 5 to arrive at the stated resut.

27 Long-time energy distribution in FPU attices Amost-invariant energies and diagona moduation functions. The foowing emma shows in particuar that z ± τ is bounded in terms of the amostinvariant E. Theorem 4. Under the conditions of Theorem, there exists c > 0 independent of E and N such that dz E zτ, dτ τ 1 cδ 4N z τ + z τ ϑ and dz E zτ, dτ τ 1 + cδ 4N z τ + z τ + ϑ for τ 1, with ϑ bounded as in 80. Proof. E has the four terms z ± ± in the sum 77 and further terms containing zj k for j, k K with k ± j and k 0. For such j, k we note that ω j k ω k ω is bounded independenty of N, and from Lemmas 8 and 10 we have the bound z k j τ d + dτ zk j τ Cδ z τ + z τ + ϑ k j + ϑ with ϑ k j bounded as in 63, and with ϑ bounded as in 68. This yieds the resut Bounding the mode energies by the amost-invariant energies. We are now in the position to bound the mode energies E j t = E j ut, ut of 8 in terms of the amost-invariants dz E t = E zτ, dτ τ for τ = N 3 t. Theorem 5. Under the conditions of Theorem, we have N E j t C + cδ δ j E t + ϑ j for t minn 3, ɛ 1/, =1 where ϑ j is bounded as in 80 and C, c are independent of E and N. The constant C ony depends on C 0 of 78, and c depends on C 0 of 6. The sum is actuay ony over with j < K + M. Proof. We insert in ɛ j t the moduated Fourier expansion 7 for ω j u j t and u j t and use the remainder bound 8. We use Lemmas 9 and 10 to bound the moduation functions zj k in terms of the diagona functions z±, and Theorem 4 to bound the diagona functions in terms of the amost-invariants E. This yieds the stated resut. Theorem 6. Under the conditions of Theorem, we have E 1 t E 1 t C E δ for t minn 3, ɛ 1/. Proof. We insert in ɛ 1 t the moduated Fourier expansion 7. Using the estimate 5 we obtain E 1 t = 4N z 1 1 τ + z 1 1 τ + OE δ. Together with Theorem 4 this gives the resut.

28 8 E. Hairer and Ch. Lubich 8.6. Dependence of the amost-invariant energies on the initia vaues. Lemma 11. In the situation of Theorem, consider perturbed initia vaues ũ0, ũ0 whose difference to u0, u0 is bounded in the norm 4 by u0 ũ0, u0 ũ0 ϑ with ϑ C δ K+M+1 N. Then, the difference of the amost-invariant energies of the associated moduation functions zj k and zk j is bounded by dz E zτ, dτ τ d z E zτ, dτ τ C ϑ δ N 1 for = 1,..., K N E dz σ zτ, dτ τ d z E zτ, dτ τ C ϑ δ K N 1 =K for τ 1, with a constant C that is independent of E and N. Proof. We foow the ines of the proof of 5, taking differences in the recursions instead of direct bounds. Omitting the detais, we obtain σ j k K j z k j z k j τ C ϑ for τ 1, with a constant C that is independent of E and N. Together with the definition of E and the bounds 5, this yieds the resut From short to ong time intervas. By the estimates of the moduation functions we have for the amost-invariants at the initia time the estimates 78, where C 0 can be chosen to depend ony on the constant C 0 of 6. We appy Theorem 3 repeatedy on intervas of ength 1. As ong as the soution ut of 3 satisfies the smaness condition 6 with a arger constant Ĉ0 in pace of C 0, Theorem gives a moduated Fourier expansion corresponding to starting vaues ut n, ut n at t n = n. We denote the sequence of moduation functions of this expansion by z n τ. The estimate 8 of Theorem for t = 1 aows us to appy Lemma 11 with ϑ Ĉ1δ K+M+1 N 4 with Ĉ1 depending on Ĉ0 to obtain, for τ = N 3, E zn τ, dzn dτ τ E zn+1 0, dz n+1 0 Ĉ δ +K+M+1 N 5 dτ for = 1,..., K, N E σ zn τ, dzn dτ τ E zn+1 0, dz n+1 0 Ĉ δ K+M+1 N 5 dτ =K with Ĉ depending on Ĉ0. Theorem 3 now yieds the same estimates with τ = 0, possiby with a different constant Ĉ depending on Ĉ0. Summing up we obtain E zn 0, dzn dτ 0 E z0 0, dz 0 dτ 0 Ĉ δ +K+M+1 N 5 t n N =K for = 1,..., K, σ E zn 0, dzn dτ 0 E z0 0, dz 0 dτ 0 Ĉ δ K+M+1 N 5 t n,

29 Long-time energy distribution in FPU attices 9 and the same estimates hod when the argument 0 of z n is repaced by τ N 3. Again Ĉ may be different and depends on Ĉ0. For t n c 0 N δ M 1 with c 0 = C 0 /Ĉ, the first expression is smaer than C 0δ +K N 3, and the second one is smaer than C 0 δ K N 3. Hence we obtain, for n c 0 N δ M 1 and τ N 3, E zn τ, dzn dτ τ C0 δ N 3 for = 1,..., K, N =K σ E zn τ, dzn dτ τ C0 δ K N 3. By Theorem 5 we therefore obtain, for t c 0 N δ M 1, E j t C + cδ δ j N 3 C δ j N 3 for = 1,..., K, N σ j E j t C + cδ δ K N 3 C δ K N 3, j=k where C ony depends on C 0 and hence on C 0, but not on Ĉ0. Provided that Ĉ 0 has been chosen such that Ĉ0 C, we see that the soution satisfies the smaness condition 6 up to times t c 0 N δ M 1, so that the construction of the moduated Fourier expansions on each of the subintervas of ength 1 is indeed feasibe with bounds that hod uniformy in n. The proof of Theorem 1 is thus compete. 9. Incuding the first near-resonance The non-resonance estimates of Section 5 are crucia for the construction of the moduated Fourier expansion. The restriction max j, µk 10 in Lemma 1 eads to the rather severe restriction K + M 10 in Theorem 1, which together with the condition K < M yieds M 4 and hence imits the resut to a time scae t N δ 5. We discuss the case K = 6, M = 5 so that K + M = 11 in order to stretch the time interva by a further factor δ 1. A difficuty now arises in the construction of zj k with j = 5 and the mutiindex k = 3, 0, 0,, 0, 0,..., 0 K j, because here ω j k ω = ω 5 ω 4 +3ω 1 = ON 5 ; see Lemma 1. We introduce the owest-order resonance modue M = {n 3, 0, 0,, 1, 0,..., 0 n Z}, which is incuded in the arger resonance modue M { k Z N N =1 k = 0, N =1 } 3 k = 0. We remove from the moduated Fourier expansion pairs j, k for which k j M or k + j M. To keep the defect sma, we have to modify the definition of the diagona coefficient functions z ± j j. We consider the pair j, k with j = 5 and k = 3, 0, 0,, 0,..., for which k j M. The use of 33 woud ead to a sma denominator of size ON 6,