A KAM theorem for Hamiltonian networks with long ranged couplings
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS JIANSHENG GENG AND YINGFEI YI
Abstract. We consider Hamiltonian networks of long ranged and weakly coupled oscillators with variable frequencies. By deriving an abstract infinite dimensional KAM type of theorem, we show that for any given positive integer N and a fixed, positive measure set O of N variable frequencies, there is a subset O∗ ⊂ O of positive measure such that each ω ∈ O∗ corresponds to a small amplitude, quasi-periodic breather (i.e., a solution which is quasi-periodic in time and exponentially localized in space) of the Hamiltonian network with N -frequencies which are slightly deformed from ω.
1. Introduction and Main Result P Associated with the symplectic structure n pn ∧ qn , we consider Hamiltonian networks defined by real analytic Hamiltonians of the form X p2 (1.1) H= ( n + Vn (qn )) + W ({qn }), 2 n∈Z
β2
where Vn ’s are the on-site potentials satisfying Vn (0) = Vn0 (0) = 0 and Vn00 (0) ≡ 2n , βn > 0, and W is a coupling potential. Hamiltonian networks have been used in solid state physics in describing the vibration of particles (atoms) in a lattice (see [10, 11, 20]) and also used to model DNA chains (see [10, 14, 34]). They also arise naturally as spatial discretization of Hamiltonian PDEs such as nonlinear wave equations. Among the solutions of a Hamiltonian network, of particular physical interests are the socalled breathers or quasi-periodic breathers, which are self-localized, time periodic or quasi-periodic, solutions whose amplitudes decay at least exponentially as |n| → ∞. Breathers or quasi-periodic breathers are often referred to as dynamical solitons or intrinsic localized modes in physics and they have been largely found via numerics in many physical models (see [10, 29] and references therein). The existence of breathers in Hamiltonian networks associated with Hamiltonians (1.1) was rigorously analyzed when βn ≡ β by Aubry [1, 2], Mackay–Aubry [25] for the inter-particle, nearest neighbor coupling potential X W ({qn }) = (qn+1 − qn )2 n
and by Bambusi [3] for the long–range coupling potential X 1 W ({qn }) = (qn − qm )2 , α > 1. |n − m|α n6=m
Like in [25], breathers in the nearest neighbor coupling case can be studied near a fixed periodic orbit of the uncoupled Hamiltonian by certain continuation or perturbation arguments, provided that the couplings are “weak”, and, no small divisors need to be considered in such perturbation problems. These perturbation techniques are also applicable in finding quasi-periodic breathers with 1991 Mathematics Subject Classification. Primary 37K60, 37K55. Key words and phrases. Coupled oscillators, Hamiltonian networks, long ranged coupling, KAM theory, quasiperiodic breathers. The second author is Partially supported by NSF grant DMS0204119. 1
2
JIANSHENG GENG AND YINGFEI YI
two or three frequencies for certain models with symmetries (see Bambusi–Vella [4], Johansson– Aubry [19]). Using a modified KAM technique, the existence of quasi-periodic breathers with any finite number of frequencies was recently shown by Yuan [33] for the higher order, nearest-neighbor coupling potential X (1.2) W ({qn }) = (qn+1 − qn )3 . n
Almost periodic breathers with infinitely many frequencies have also been investigated. Associated with the potential (1.2), Fr¨ ohlich-Spencer-Wayne [15] considered the case when the frequencies are non-negative random variables with smooth distribution of fast decay at infinity and showed that there is a set Ω ⊂ R∞ + with positive probability measure such that each ω ∈ Ω corresponds to an almost periodic breather with infinite many frequencies (see also P¨oschel [28] for more general spatial structures). In this paper, we will study the existence of quasi-periodic breathers for the Hamiltonian (1.1) with the following higher order, long-ranged coupling potential 1 X −|n−m|α e (qn − qm )3 , α ≥ 1, (1.3) W ({qn }) = 3 n6=m
or equivalently the Hamiltonian network X α d 2 qn + Vn0 (qn ) = − e−|n−m| (qn − qm )2 . 2 dt
(1.4)
m∈Z
For a given integer N > 1, we specify N integers {i1 , · · · , iN } and let Z1 = Z \ {i1 , · · · , iN }. N and assume the We treat ω = (βi1 , · · · , βiN ) as parameters in a bounded closed region O in R+ following spectral gap condition: SG) There exist 1 ≤ d < ∞ and γ > 0 such that {βn }n∈Z1 = ∪∞ l=1 Λl where Λl , l = 1, 2, · · · , are sets satisfying #(Λl ) ≤ d,
for all l,
and |βn − βm | ≥ γ,
for all βn ∈ Λl , βm ∈ Λj , l 6= j.
We will show the following result. Theorem A. Assume SG) with γ sufficiently small. Then there exists a Cantor set Oγ ⊂ O, with meas(O \ Oγ ) = O(γ), such that for any ω ∈ Oγ , the Hamiltonian network (1.4) associated with ω admits a small amplitude, linearly stable, quasi-periodic breather q(t) = ({qn (t)}) of N -frequency ω∗ which is close to ω, and moreover, |qn | ∼ e−|n| . The condition SG) clearly holds when βn = |n|, n ∈ Z1 . Comparing with the cases of nonlinear wave equations [12, 17, 26, 27] in which βn ∼ |n|, n ∈ Z, the validity of Theorem A crucially depends on the coupling potential or perturbation (1.3) which admits a weaker regularity but a higher order perturbation. For the case of nearest-neighboring coupled Hamiltonian networks, breathers were shown to be super-exponentially localized in space ([33]). This is due to the at most linear growth of the normal components in the normal form associated with the short-ranged coupling potential (1.2). Our result only asserts the exponential localization of quasi-periodic breathers due to the exponential growth of the normal components in the normal form associated with the exponentially weighted, long-ranged coupling potential (1.3). If the long-ranged coupling potential 1 1 X (qn − qm )3 , α > 1 W ({qn }) = 3 |n − m|α n6=m
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
3
is considered instead, then the normal components in the associated normal form will have a superexponential growth, and our method will equally applicable to yield quasi-periodic breathers which are localized like |n|1 α in space. Theorem A will be proved by using KAM (Kolmogorov-Arnold-Moser) method. In fact, we will present an abstract infinite dimensional KAM type of theorem from which Theorem A will follow. Such an infinite dimensional KAM theorem differs significantly from those for Hamiltonian PDEs like nonlinear Sch¨ odinger, wave, beam, and KdV equations studied by many authors using either KAM or CWB (Craig-Wayne-Bourgain) method (see [5, 6, 7, 8, 9, 12, 13, 16, 17, 18, 21, 22, 23, 26, 27, 30] and references therein). This is mainly due to the fact that, when the normal frequencies of a Hamiltonian network have linear growth, the perturbation (1.3) admits weaker regularity than those of Hamiltonian PDEs under KAM or Newton iterations. Similar to the short-ranged coupling cases considered in [25, 33], it is also important to study the existence of quasi-periodic breathers for Hamiltonian networks with long-ranged coupling potentials and constant frequencies βn ≡ β, n ∈ Z, i.e., those formed by weakly coupled identical oscillators. As the KAM iteration mechanism and measure estimates for the constant frequency case significantly differ from the variable ones to be studied in this paper, we will consider the constant frequency case in a separate work. The paper is organized as follows. In Section 2 we state an abstract infinite dimensional KAM theorem and prove Theorem A as a corollary. Sections 3 and 4 are devoted to the proof of the abstract KAM theorem. More precisely, in Section 3, we give detailed construction of the KAM iteration for one KAM step. We complete the proof of the abstract infinite dimensional KAM theorem in Section 4 by showing an iteration lemma, convergence, and measure estimate. Some technical lemmas are provided in the Appendix. 2. An Abstract KAM Theorem In this section, we will formulate an abstract KAM theorem which can be applied to the Hamiltonian networks of long-ranged and weakly coupled oscillators with variable frequencies. Theorem A will be proved by using the abstract KAM theorem and normal form reductions. 2.1. The abstract theorem. We begin with some notations. Let integers N > 1, d ≥ 1, and real numbers r, s > 0 be given. We consider the complex neighborhood D(r, s) of TN ×{0}×{0}×{0} ⊂ TN × RN × `1 × `1 defined by D(r, s) = {(θ, I, w, w) ¯ : |Imθ| < r, |I| < s2 , kwk < s, kwk ¯ < s}, where | · | denote the sup-norm of complex vectors and k · k denote the `1 norm. Also let O be a positive (Lebesgue) measure set in RN . Let F (θ, I, w, w) ¯ be a real analytic function on D(r, s) which depends on a parameter ξ ∈ O, 2 2 C d -Whitney smoothly (i.e., C d in the sense of Whitney). We expand F into the Taylor-Fourier series with respect to θ, I, w, w: ¯ X F (θ, I, w, w) ¯ = Fαβ wα w ¯β , α,β
where α ≡ (· · · , αn , · · · ), β ≡ (· · · , βn , · · · ), αn , βn ∈ N, are multi-indices with finitely many non-vanishing components, and X Fαβ = Fklαβ (ξ)I l eihk,θi . k∈ZN ,l∈NN
We define the weighted norm of F by kF kD(r,s),O ≡ sup kwk<s kwk<s ¯
X α,β
kFαβ k |wα ||w ¯ β |,
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JIANSHENG GENG AND YINGFEI YI
where kFαβ k ≡
X
|Fklαβ |O s2|l| e|k|r ,
|Fklαβ |O ≡ sup { max2 |∂ξm Fklαβ |}. ξ∈O m≤d
k,l
In the above and also for the rest of the paper, derivatives in ξ ∈ O are taken in the sense of Whitney. For a vector-valued function G : D(r, s) × O → Cn , n < ∞, we simply define its weighed norm by n X kGkD(r,s),O ≡ kGi kD(r,s),O . i=1
For the Hamiltonian vector field XF = (FI , −Fθ , {iFwn }, {−iFw¯n }) associated with a Hamiltonian function F on D(r, s) × O, we define its weighted norm by X 1 X 1 kFwn kD(r,s),O + kFw¯n kD(r,s),O ). kXF kD(r,s),O ≡ kFI kD(r,s),O + 2 kFθ kD(r,s),O + ( s s n n X Associated with the symplectic structure dI ∧ dθ + i dwn ∧ dw ¯n , we consider the following n∈Z
family of real analytic, parameterized Hamiltonians (2.1)
H
= N + P,
N
= hω(ξ), Ii +
X
Ωn wn w ¯n ,
n∈Z
P
= P (θ, I, w, w, ¯ ξ),
where (I, θ, w, w) ¯ ∈ D(r, s), ξ ∈ O, Ωn ’s are positive and independent of ξ, and all ξ-dependence 2 are of class C d in the sense of Whitney. It is clear that when P = 0, the unperturbed Hamiltonians N are completely integrable, admitting a family of quasi-periodic solutions (θ + ωt, 0, 0, 0) corresponding to invariant N -tori in the phase space. To study the persistence of some of these N -tori, we need the following assumptions on ω(ξ), Ωn and the perturbation P : (A1) Non-degeneracy of tangential frequencies: There is a constant δ > 0 such that ∂ω | det ( )| ≥ δ. ∂ξ (A2) Gap conditions of normal frequencies: There exist sufficiently small γ > 0 and sets Λl , l = 1, 2, · · · , such that {Ωn }n∈Z = ∪∞ l=1 Λl , #(Λl ) ≤ d, for all l, |Ωn − Ωm | ≥ γ, if Ωn ∈ Λl , Ωm ∈ Λj , l 6= j.
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
5
(A3) Decay property of the perturbation: P = P˘ + P´ + P` , where P˘ = P˘ (θ, I, w, w, ¯ ξ), P´ = ´ ` ` P (θ, I, w, w, ¯ ξ), P = P (θ, I, w, w, ¯ ξ) are such that X (2.2) P˘ = P˘ (θ, I, 0, 0, ξ) + P˘n (θ, I, ξ)wnαn w ¯nβn , kP˘n (θ, I, ξ)k ≤ e−|n| ; n∈Z αn +βn ≥1
(2.3)
P´
X
=
αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m ,
kP´nm (ξ)k ≤ e−|n−m| ;
n,m∈Z,n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
(2.4)
P`
=
X
O(|wn |3 ).
n∈Z
Our abstract KAM theorem states as the following. Theorem B. Consider the Hamiltonian (2.1) and assume (A1)-(A3). For a fixed γ > 0 sufficiently small, there exists a positive constant ε = ε(O, d, δ, N, γ, r, s) such that if kXP kD(r,s),O < ε, then the following holds. There exist Cantor sets Oγ ⊂ O with meas(O \ Oγ ) = O(γ) and maps Ψ : TN × Oγ → D(r, s),
ω ˜ : Oγ → R N ,
2
which are real analytic in θ and C d -Whitney smooth in ξ with kΨ − Ψ0 kD( r2 ,0),Oγ → 0 and |˜ ω − ω|Oγ → 0 as γ → 0, where Ψ0 is the trivial embedding: TN × O → TN × {0, 0, 0}, such that each ξ ∈ Oγ and θ ∈ TN corresponds to a linearly stable, N -frequency quasi-periodic solution Ψ(θ + ω ˜ (ξ)t, ξ) = (θ + ω ˜ (ξ)t, I(t), {wn (t)}) of the Hamiltonian (2.1). Moreover, |wn | ∼ e−|n| . Since our perturbation has a weaker regularity, the frequencies of these invariant tori are in general non-resonant instead of Diophantine. Comparing with results on quasi-periodic solutions for Hamiltonian PDEs (see e.g. [5, 6, 7, 8, 9, 12, 13, 16, 17, 18, 21, 22, 23, 26, 27, 30]), the above theorem relaxes the linear or super-linear growth conditions on the normal frequencies Ωn . Indeed, it is easy to see that the gap condition (A2) above is weaker than the linear or sup-linear growth conditions on the normal frequencies. The assumption (A3) is new but natural for networks of long ranged and weakly coupled oscillators. It is not clear whether a Lyapunov center theorem is possible for a Hamiltonian network whose normal frequencies satisfy the gap condition (A2). At least, the above theorem assert a quasiperiodic type of Lyapunov center result in the sense of measure. 2.2. Proof of Theorem A. Recall that the Hamiltonian networks of long ranged, weakly coupled oscillators considered in Theorem A is described by the Hamiltonian X p2 1 X −|n−m|α H= [ n + Vn (qn )] + e (qn − qm )3 , α ≥ 1, 2 3 n∈Z
n6=m
which, in terms of the Taylor expansion at q = 0, can be equivalently rewritten as X p2 X β 2 q2 1 X −|n−m|α H= [ n + n n] + e (qn − qm )3 + O(|qn |3 ). 2 2 3 n∈Z
n∈Z
n6=m
Let ε > 0 be sufficiently small. With the re-scalings pn , qn → εpn , εqn , the re-scaled Hamiltonian reads X p2 X β 2 q2 ε X −|n−m|α ε−2 H(εp, εq) = [ n + n n] + e (qn − qm )3 + ε O(|qn |3 ). 2 2 3 n∈Z
n6=m
n∈Z
Let N , {i1 , · · · , iN }, and Z1 = Z \ {i1 , · · · , iN } be as in Theorem A. For a given value a = (a1 , · · · , aN ) ∈ RN ¯ = + , we introduce the standard action-angle-normal variables (I, θ, w, w)
6
JIANSHENG GENG AND YINGFEI YI
(I, θ, {wn }n∈Z1 , {w ¯n }n∈Z1 ) ∈ RN × TN × `1 , i.e., s q p 1 p pij = βij Ij + aj cos θj , qij = Ij + aj sin θj , 1 ≤ j ≤ N, βij √ βn (wn + w ¯n ) wn − w ¯n √ , n ∈ Z1 . pn = , qn = √ i 2βn 2 Let ξ = (ξ1 , · · · , ξN ) = (βi1 , · · · , βiN ). Then in terms of the action-angle-normal variables the above Hamiltonian becomes X Ωn wn w ¯n + P (θ, I, w, w, ¯ ξ), (2.5) H = N + P = hω(ξ), Ii + n∈Z1
where ω(ξ) = (ω1 (ξ), · · · , ωN (ξ)) = (ξ1 , · · · , ξN ), Ωn = βn , n ∈ Z1 , and P = P˘ + P´ + P` satisfying P˘
= P˘ (θ, I, 0, 0, ξ) +
X
P˘n (θ, I, ξ)wnαn w ¯nβn ,
α
kP˘n (θ, I, ξ)k ≤ e−|n| ≤ e−|n| ;
n∈Z1 αn +βn ≥1
P´
X
=
αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m ,
α
kP´nm (ξ)k ≤ e−|n−m| ≤ e−|n−m| ;
n,m∈Z1 ,n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
P`
=
X
O(|wn |3 ).
n∈Z1
It is also easy to see that we can choose appropriate r, s > 0 such that kXP kD(r,s),O < ε. Hence the Hamiltonian (2.5) satisfies all conditions of Theorem B, from which Theorem A follows. 3. KAM Step In what follows, we will perform KAM iterations to (2.1) which involves infinite many successive steps, called KAM steps, of iterations, to eliminate lower order θ-dependent terms in P . Each KAM step will make the perturbation smaller than the previous one at a cost of excluding a small measure set of parameters. At the end, the KAM iterations will be shown to converge and the measure of the total excluding set will be shown to be small. To begin with the KAM iteration, we set Ω0n = Ωn , n ∈ Z1 , r0 = r, s0 = s. 3.1. Normal form. We first convert the Hamiltonian (2.1) into a more convenient form in order 5 to perform the KAM iteration. Let ε∗ ∼ ε 4 and choose a K0 such that K0 ∼ | ln ε∗ |. According to the forms of (2.3), (2.4) in the Assumption (A3), we can make s0 smaller if necessary such that kXP´ +P` kD(r0 ,s0 )×O ≤ ε∗ . We now treat the term P˘ . According to the form of (2.2) and the definition of the norm, we have X P˘ = P˘ (θ, I, 0, 0, ξ) + P˘n (θ, I, ξ)wnαn w ¯nβn αn +βn ≥1
=
X
P˘kl I l eihk,θi +
k,l
X
P˘nklαn βn I l eihk,θi wnαn w ¯nβn ,
n,k,l αn +βn ≥1
where kP˘kl k ≤ e−|k|r0 ,
kP˘nklαn βn k ≤ e−|k|r0 e−|n| .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
7
Thus we can make r0 , s0 smaller such that kXP
|k|>K0 ,|l|≤1
P P˘kl I l eihk,θi + |n|>K0
or |k|>K0 αn +βn ≥1
αn βn P˘nkαn βn eihk,θi wn w ¯n +O(|I|2 )+O(|w|3 ) k
≤ ε∗ .
Let R=
X
X
P˘kl I l eihk,θi +
P˘nkαn βn eihk,θi wnαn w ¯nβn .
|n|≤K0 ,|k|≤K0 1≤αn +βn ≤2
|k|≤K0 ,|l|≤1
We first construct a symplectic transformation Φ∗ = Φ1F∗ defined as the time-1 map of the Hamiltonian flow associated to a Hamiltonian F∗ of the form X X F∗ = Fkl00 I l eihk,θi + (Fnk10 wn + Fnk01 w ¯n )eihk,θi |k|≤K0 ,|n|≤K0
0K0 αn +βn ≥1
satisfies kP˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)k ≤ e−(|n|−K0 ) . In the above, the first and the second term of P˘∗ come from P ◦ Φ∗ and P˘ ◦ Φ∗ + P´ ◦ Φ∗ respectively. The decay property of P˘n∗ follows from the fact that Φ∗ depends only on I, θ and wm , w ¯m for |m| ≤ K0 .
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JIANSHENG GENG AND YINGFEI YI
Next, write X
P˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)wnαn w ¯nβn
|n|>K0 αn +βn ≥1
=
X
P˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)wnαn w ¯nβn
|n|>5K0 αn +βn ≥1
+
X
¯nβn . P˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)wnαn w
K0 K αn +βn ≥1
= P˘ ν (θ, I, z ν , z¯ν , ξ) +
X
P˘nν (θ, I, z ν , z¯ν , ξ)wnαn w ¯nβn
˜ν |n|>K αn +βn ≥1
˜ =K ˜ ν is a positive constant, L = Lν defined on a domain D(r, s) × O = D(rν , sν ) × Oν , where K 0 is the smallest positive integer such that {Ωn }|n|≤K˜ lie in Λ1 , · · · , ΛL , zl = zlν = (· · · , wn , · · · ) Ωn ∈Λl , ˜ |n|≤K
ν
z = z = (z1 , · · · , zL ), P = P ν , for some ε = εν , and
z¯l = z¯lν = (· · · , w ¯n , · · · ) Ωn ∈Λl , ˜ |n|≤K
ν
z¯ = z¯ = (¯ z1 , · · · , z¯L ), ˜
kP˘n (θ, I, z, z¯, ξ)kD(r,s),O ≤ e−(|n|−K) ,
˜ |n| > K.
˜ It is clear that L ≤ K. We will construct a symplectic transformation Φ = Φν , which, in smaller frequency and phase domains, carries the above Hamiltonian into the next KAM cycle. Below, all constants c1 − c12
10
JIANSHENG GENG AND YINGFEI YI
are positive and independent of the iteration process. The tensor product (or direct product) of two m × n, k × l matrices A = (aij ), B is a (mk) × (nl) matrix defined by a11 B · · · a1n B ··· ··· . A ⊗ B = (aij B) = · · · am1 B · · · amn B We also use k · k to denote the operator matrix norm, i.e., for a matrix M , kM k = supkyk=1 kM yk. ˜ < |n| ≤ K ˜ + will be added to the new ˜+ = 7 K ˜ + 1 K0 . In this KAM step, Ω0n with K Let K 4 4 normal matrix A+ according to the assumption (A2). In order to have a compact formulation l when solving a homological equation, we rewrite N as N
e + hω(ξ), Ii +
=
L X
hAl zl , z¯l i
l=1
X
+
X
Ω0n wn w ¯n +
Ω0n wn w ¯n
˜+ |n|>K
˜ ˜+ KK
l=1
where dim(A˜l ) ≤ d, L+ is the smallest positive integer such that {Ωn }|n|≤K˜ + lie in Λ1 , · · · , ΛL+ ˜ + ), (hence L+ ≤ K Al 0 ˜ < |n| ≤ K ˜ +, A˜l = , Ω0n ∈ Λl , K 0 Ω0n zl+ = (· · · , wn , · · · )
Ωn ∈Λl ˜ |n|≤K +
,
+ z + = (z1+ , · · · , zL ), +
z¯l+ = (· · · , w ¯n , · · · )
Ωn ∈Λl ˜ |n|≤K +
,
+ z¯+ = (¯ z1+ , · · · , z¯L ). +
3.2. Truncation. We first expand P˘ into the Taylor-Fourier series X X ¯nβn , P˘ = P˘klαβ eihk,θi I l z α z¯β + P˘klnαβ eihk,θi I l z α z¯β wnαn w k,l,α,β
k,l,n,α,β ˜ |n|>K,α n +βn ≥1
where k ∈ ZN , l ∈ NN and the multi-index α (resp. β) runs over the set α ≡ (α1 , · · · , αl , · · · , αL ) for αl = (· · · , αm , · · · ) Ωm ∈Λl , αm ∈ N (resp. β ≡ (β 1 , · · · , β l , · · · , β L ) for β l = (· · · , βm , · · · ) Ωm ∈Λl , ˜ |m|≤K
˜ |m|≤K
βm ∈ N). Let R be the following truncation of P˘ : X
R(θ, I, z, z¯, w, w) ¯ =
P˘kl00 eihk,θi I l
|k|≤K+ ,|l|≤1
+ +
X
(hP˘lk10 , zl i |k|≤K+ ,l≤L X
+ hP˘lk01 , z¯l i)eihk,θi +
X
(P˘nk10 wn + P˘nk01 w ¯n )eihk,θi
˜ ˜+ |k|≤K+ ,KK 1
where h.o.t. denotes the terms of the form O(|I|2 + |I||w| + |w|3 ). Let r+ = 2r + r40 , η = ε 4 . Using the facts ˜ kP˘n (θ, I, z, z¯, ξ)k ≤ e−(|n|−K) , kP˘k (I, z, z¯, ξ)k ≤ e−|k|r , we have that if r−r+ 5 C1) e−K+ 2 ≤ ε 4 ,
12
JIANSHENG GENG AND YINGFEI YI
then (3.3)
kXP −R kD(r
r−r+ ++ 2
,ηs),O
X
≤
e−|k|
r−r+ 2
˜
X
+
5
e−(|n|−K) + h.o.t. ≤ c1 ε 4 .
˜+ |n|>K
|k|>K+
3.3. The homological equation. We now look for a Hamiltonian F , defined in a domain D+ = D(r+ , s+ ), such that the time-1 map Φ = Φ1F of the Hamiltonian vector field XF defines a map from D+ to D and transforms H into H+ in the next KAM cycle. Let F have the form F (θ, I, z + , z¯+ )
= F0 + F1 + F2 X = Fkl00 eihk,θi I l +
(fnk10 wn + fnk01 w ¯n )eihk,θi
˜+ |k|≤K+ ,|n|≤K
0 . Similarly, Fljk11 = (Fjlk11 )> and Fljk02 = (Fjlk02 )> . Hence, the Hamiltonian F is uniquely determined on O+ . We proceed to estimate XF and Φ1F . . Lemma 3.3. Let Di = D(r+ + 4i (r − r+ ), 4i s), 0 < i ≤ 4. If 4
1
τd C2) K+ ≤ ε− 4 , then there is a constant c2 > 0 such that 4
3
kXF kD3 ,O+ ≤ c2 γ −d (r − r+ )−N ε 4 . Proof. By the definition of O+ , Lemma 3.1, Lemma 3.2 and Lemma 5.5, Lemma 5.6 in the Appendix, we have 2 4 τ d4 ˘ |Fkl00 |O+ ≤ |hk, ωi|−d |P˘kl00 |O+ ≤ γ −d K+ |Pkl00 |O+ , k 6= 0; 4
4
4
4
4
4
4
4
4
4
kFlk10 kO+
τd ≤ γ −d K+ kRlk10 kO+ ;
kFlk01 kO+
τd ≤ γ −d K+ kRlk01 kO+ ;
kFjlk20 kO+
τd k20 ≤ γ −d K+ kRjl kO+ ;
kFjlk11 kO+
τd k11 ≤ γ −d K+ kRjl kO+ , |k| + |l − j| = 6 0;
kFjlk02 kO+
τd k02 ≤ γ −d K+ kRjl kO+ .
It follows that 1 kFθ kD3 ,O+ s2
1 1 X ( |Fkl00 | · s2|l| · |k| · e|k|(r− 4 (r−r+ )) 2 s k,|l|≤1 X 1 + (kFlk10 k · kzl+ k) · |k| · e|k|(r− 4 (r−r+ ))
≤
k,l
+
X
+
X
1
(kFlk01 k · k¯ zl+ k) · |k| · e|k|(r− 4 (r−r+ ))
k,l 1
kFjlk20 k · kzl+ kkzj+ k · |k| · e|k|(r− 4 (r−r+ ))
k,l,j
+
X
1
kFjlk11 k · kzl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ ))
|k|+|l−j|6=0
+
X k,l,j
1
kFjlk02 k · k¯ zl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ )) )
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
15
4 τ d4 X γ −d K+ 1 ( |P˘kl00 | · s2|l| · |k| · e|k|(r− 4 (r−r+ )) 2 s k,|l|≤1 X 1 + (kRlk10 k · kzl+ k) · |k| · e|k|(r− 4 (r−r+ ))
≤
k,l
+
X
+
X
1
(kRlk01 k · k¯ zl+ k) · |k| · e|k|(r− 4 (r−r+ ))
k,l 1
k20 kRjl k · kzl+ kkzj+ k · |k| · e|k|(r− 4 (r−r+ ))
k,l,j 1
X
+
k11 kRjl k · kzl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ ))
|k|+|l−j|6=0
+
X
1
k02 kRjl k · k¯ zl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ )) )
k,l,j 4
4
τd ≤ c3 γ −d (r − r+ )−N K+ kXR k 4
3
≤ c3 γ −d (r − r+ )−N ε 4 . Similarly, kFI kD3 ,O+
=
X
4
1
3
|Fkl00 |e|k|(r− 4 (r−r+ )) ≤ c4 γ −d (r − r+ )−N ε 4 .
|l|=1
kXF1 kD3 ,O+
≤
X X 1 X 1 X ( kF1wn k + kF1w¯n k) ≤ ( kF1z+ k + kF1z¯+ k) s n s l l n l
kXF2 kD3 ,O+
−d4
−N
τ d4 K+ kXR1 k
−d4
−N
τ d4 K+ kXR2 k
l
−d4
−N
−d4
−N
3
≤ c5 γ (r − r+ ) ≤ c5 γ (r − r+ ) ε 4 . X X 1 X 1 X ( kF2z+ k + kF2z¯+ k) ≤ kF2wn k + kF2w¯n k) ≤ ( s n s l l n l
≤ c6 γ
(r − r+ )
≤ c6 γ
l
(r − r+ )
3
ε4 .
The proof is now completed by adding the estimates above together. Let Diη = D(r+ + 4i (r − r+ ), 4i ηs), 0 < i ≤ 4.
Lemma 3.4. If 4 1 C3) c2 γ −d (r − r+ )−N ε 2 < 1, then ΦtF : D2η → D3η ,
(3.9)
−1 ≤ t ≤ 1,
and moreover, 4
3
kDΦtF − IkD1η < c7 γ −d (r − r+ )−N ε 4 .
(3.10) Proof. Let
kDm F kD,O+ = max{k
∂ |i|+|l|+|α|+|β| ∂θi ∂I l ∂(z + )α ∂(¯ z + )β
F kD,O+ , |i| + |l| + |α| + |β| = m ≥ 2}.
We note that F is a polynomial of order 1 in I and of order 2 in z + , z¯+ . It follows from Lemma 3.3 and the Cauchy inequality that 4
3
kDm F kD2 ,O+ < c8 γ −d (r − r+ )−N ε 4 , for any m ≥ 2. Using the integral equation ΦtF
Z = id + 0
t
XF ◦ ΦsF ds
16
JIANSHENG GENG AND YINGFEI YI
and Lemma 3.3, we easily see that ΦtF : D2η → D3η , −1 ≤ t ≤ 1. Since Z t Z t DΦtF = Id + (DXF )DΦsF ds = Id + J(D2 F )DΦsF ds, 0
0
where J denotes the standard symplectic matrix. Let c7 = 2c8 . It follows that 4
3
kDΦtF − Ik ≤ 2kD2 F k ≤ c7 γ −d (r − r+ )−N ε 4 . 3.4. The new Hamiltonian. Let Φ = Φ1F , s+ = 18 ηs, D+ = D(r+ , s+ ), and = e+ + hω+ , Ii +
N+
L+ X
X
+ + hA+ ¯l i + l zl , z
Ω0n wn w ¯n ,
˜+ |n|>K
l=1
= P˘+ + P´0 + P`0 ,
P+ where e+ A+ l zl+ +
z
P˘+
= e + P˘0000 , ω+ = ω + P˘0l00 (|l| = 1), = A˜l + R011 , l ≤ L+ , ll
=
(· · · , wn , · · · )|n|≤K˜ + ,
z¯l+ = (· · · , w ¯n , · · · )|n|≤K˜ + ,
+ + (z1+ , · · · , zL ), z¯+ = (¯ z1+ , · · · , z¯L ), + + Z 1 Z 1 Z 1 = (1 − t){{N, F }, F } ◦ ΦtF dt + {R, F } ◦ ΦtF dt + (P˘ − R) ◦ Φ1F + {P´0 + P`0 , F } ◦ ΦtF dt.
=
0
0
0
Then Φ : D+ × O+ → D, and, by the second order Taylor formula, H+
≡ H ◦ Φ = (N + R) ◦ Φ + (P − R) ◦ Φ Z 1 = N + {N, F } + R + (1 − t){{N, F }, F } ◦ ΦtF dt 0 1
Z
{R, F } ◦ ΦtF dt + (P˘ − R) ◦ Φ1F + (P´0 + P`0 ) ◦ Φ1F
+ 0
= N + {N, F } + R + P˘+ + P´0 + P`0 = N+ + P + + {N, F } + R − P˘0000 − hω 0 , Ii −
L+ X
011 + + hRll zl , z¯l i
l=1 +
= N+ + P . Below, we show that the new Hamiltonian H+ enjoys similar properties as H. By the Assumptions of P˘ , we have that there is a constant c9 > 0 such that |ω+ − ω|O ≤ c9 ε, kA+ − A˜l kO ≤ c9 ε. +
l
+
Denote R(t) = (1 − t)(N+ − N ) + tR. We can rewrite P + as Z 1 Z 1 + t P = (1 − t){{N, F }, F } ◦ ΦF dt + {R, F } ◦ ΦtF dt + (P − R) ◦ Φ1F 0
Z =
0 1
{R(t), F } ◦ ΦtF dt + (P − R) ◦ Φ1F .
0
Hence Z XP + = 0
By Lemma 3.4, if 4 3 C4) c7 γ −d (r − r+ )−N ε 4 ≤ 1,
1
(ΦtF )∗ X{R(t),F } dt + (Φ1F )∗ X(P −R) .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
17
then kDΦtF kD1η ≤ 1 + kDΦtF − IkD1η ≤ 2, By Lemma 5.4 and (3.3), we also have
−1 ≤ t ≤ 1.
4
7
kX{R(t),F } kD2η ≤ c10 γ −d (r − r+ )−N η −2 ε 4 , 5
kX(P −R) kD2η ≤ c1 ε 4 . Let c0 = max{c1 , · · · , c10 , c11 , c12 }, where c11 , c12 will be defined later, and let 4
5
ε+ = 4c0 γ −d (r − r+ )−N ε 4 . Then
4
5
5
kXP + kD+ ,O+ ≤ 2c1 ε 4 + 2c10 γ −d (r − r+ )−N ε 4 ≤ ε+ . We now exam the decay property of P˘+ . More precisely, write X P˘+ = P˘ + (θ, I, z + , z¯+ , ξ) + P˘n+ (θ, I, z + , z¯+ , ξ)wnαn w ¯nβn . ˜ + ,αn +βn ≥1 |n|>K
We show that
˜
kP˘n+ (θ, I, z + , z¯+ , ξ)kD+ ,O+ ≤ e−(|n|−K+ ) ,
|n| >˜˜K+ . ˜ + , so does {N, F }. Hence Since F only involves the normal components wn , w ¯n for |n| ≤ K R1 t ˜ + . Recall ¯n for |n| ≤ K (1 − t){{N, F }, F } ◦ ΦF dt only involves the normal components wn , w 0 that X P`0 = O(|wn |3 ). n
R ˜ + , so does 1 {P`0 , F }◦Φt dt. Hence {P`0 , F } only involves the normal components wn , w ¯n for |n| ≤ K F R1 0 Since R is a truncation of P˘ , we only need to consider the terms (P˘ − R) and 0 {P˘ + P´0 , F } ◦ ΦtF dt. Recall that X ¯nβn , P˘n (θ, I, z, z¯, ξ)wnαn w P˘ = P˘ (θ, I, z, z¯, ξ) + ˜ n +βn ≥1 |n|>K,α ˜
kP˘n (θ, I, z, z¯, ξ)kD(r,s),O+ ≤ e−(|n|−K) . X 0 αm βm w ¯m , P´0 = P´nm (ξ)wnαn w ¯nβn wm n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
0 kP´nm (ξ)kD(r,s),O+ ≤ e−|n−m| . ˜ + , the terms corresponding to the Since R only involves the normal components wn , w ¯n for |n| ≤ K ˜ ˘ ˜ + in P˘ , normal components wn , w ¯n for |n| > K+ in P − R are just those corresponding for |n| > K for which we have the decay property ˜
˜
kP˘n (θ, I, z, z¯, ξ)kD(r+ ,s+ ),O+ ≤ e−(|n|−K) ≤ e−(|n|−K+ ) . R1 To prove the decay estimates of 0 {P˘ + P´0 , F } ◦ ΦtF dt, we only need to consider the terms cor˜ + . Since F is independent of normal responding to the normal components wn , w ¯n for |n| > K R1 ˜ ˘ components wn , w ¯n for |n| > K+ , so is 0 {P (θ, I, z, z¯, ξ), F } ◦ ΦtF dt. Similarly, Z 1 X 0 αm βm { P´nm (ξ)wnαn w ¯nβn wm w ¯m , F } ◦ ΦtF dt 0
˜ n6=m,|n|,|m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
18
JIANSHENG GENG AND YINGFEI YI
˜ + . Thus, it remains to consider terms is independent of the normal components wn , w ¯n for |n| > K Z 1 X { P˘n (θ, I, z, z¯, ξ)wnαn w ¯nβn , F } ◦ ΦtF dt 0
˜ n +βn ≥1 |n|>K,α 1
Z
X
= 0
(3.11)
{P˘n (θ, I, z, z¯, ξ), F } ◦ ΦtF wnαn w ¯nβn dt
˜ n +βn ≥1 |n|>K,α
Z 1 ( {P˘n (θ, I, z, z¯, ξ), F } ◦ ΦtF dt)wnαn w ¯nβn ,
X
=
0
˜ n +βn ≥1 |n|>K,α
and 1
Z
X
{ 0
1
Z
X
= 0
(3.12)
˜+ |n|>K
X
{
˜+ |n|>K
X
=
0 αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m , F } ◦ ΦtF dt
˜ ,|m|≤K ˜ n6=m,|n|>K + + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
˜ |m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
Z 1 ( { 0
0 αm βm P´nm (ξ)wm w ¯m , F } ◦ ΦtF wnαn w ¯nβn dt
X
0 αm βm ¯nβn . w ¯m , F } ◦ ΦtF dt)wnαn w P´nm (ξ)wm
˜ |m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
Let P˜n = P˘n (θ, I, z + , z¯+ , ξ) +
X
0 P´nm (ξ).
˜ |m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
We combine (3.11) and (3.12) to consider decay property of Z 1 X ( {P˜n , F } ◦ ΦtF dt)wnαn w ¯nβn . ˜ n +βn ≥1 |n|>K,α
0
˜
˜
By relaxing decay properties of e−(|n|−K) , e−|n−m| to e−(|n|−K+ ) , we have by Lemma 5.3 that 4
3
˜
k{P˜n , F }kD(r−σ, 12 s) ≤ c11 γ −d (r − r+ )−N σ −1 s−2 ε 4 e−(|n|−K+ ) . It follows from Cauchy estimate that 4
3
˜
kX{P˜n ,F } kD(r−2σ, 14 s) ≤ c12 γ −d (r − r+ )−N σ −2 s−4 ε 4 e−(|n|−K+ ) . Hence by Lemma 3.4, if 4 3 C5) c11 γ −d (r − r+ )−N η −2 ε 4 ≤ 21 , 4
3
C6) c12 c2 (γ −d (r − r+ )−N η −2 ε 4 )2 ≤ 12 , then Z 1 k {P˜n , F } ◦ ΦtF dtkD(r+ ,s+ ) 0
k{P˜n , F } ◦ ΦtF kD(r+ ,s+ ) ≤ k{P˜n , F }kD(r+ ,s+ ) + k{P˜n , F } ◦ ΦtF − {P˜n , F }kD(r+ ,s+ ) ≤ k{P˜n , F }kD(r+ ,s+ ) + kX{P˜n ,F } kD2η kΦtF − idkD1η ≤
4
3
˜
4
3
˜
≤ c11 γ −d (r − r+ )−N η −2 ε 4 e−(|n|−K+ ) + c12 c2 (γ −d (r − r+ )−N η −2 ε 4 )2 e−(|n|−K+ ) ˜
≤ e−(|n|−K+ ) .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
This completes one step of KAM iterations.
19
4. Proof of Theorem B ˜ 0 , O0 , H0 be given at the beginning of Section 3. For each ν = 0, 1, · · · , we Let r0 , s0 , ε0 , γ, K0 , K label all index-free quantities by ν and label all +-indexed quantities by ν + 1. This defines, for all ν = 1, 2, · · · , the following sequences: rν
= r0 (1 −
ν+1 X
2−i ),
i=2 −d4
5
4 , (rν−1 − rν )−N εν−1
εν
=
4c0 γ
sν
=
ν−1 Y 1 1 1 ην−1 sν−1 = 2−3ν ( εi ) 4 s0 , ην = εν4 , 8 i=0
Kν ˜ν K
=
Dν ˜ν D Hν Nν
4Kν−1 , ˜ ν−1 + Kν , = K = D(rν , sν ), 1 1 = D(rν+1 + (rν − rν+1 ), ην sν ), 4 4 = Nν + Pν , Lν X X hAνl zlν , z¯lν i + = eν + hων (ξ), Ii + Ω0n wn w ¯n , l=1
Oν
˜ν |n|>K
τ d2 Kν −1 |hk, ω i | ≤ , 0 < |k| ≤ Kν ν−1 γ τ d2 Kν −1 ˜ν , |k| ≤ Kν , l ≤ Lν ≤ K k(hk, ων iI + A˜ν−1 ) k ≤ l γ 2 ξ ∈ Oν−1 : = τ d K k(hk, ων−1 iI + A˜lν−1 ⊗ I + I ⊗ A˜ν−1 )−1 k ≤ γν , |k| ≤ Kν , l, j ≤ Lν j 2 d |k| + |l − j| 6= 0, Kτ k(hk, ων−1 iI + A˜lν−1 ⊗ I − I ⊗ A˜ν−1 )−1 k ≤ γν , j |k| ≤ Kν , l, j ≤ Lν
where Lν is the smallest positive integer such that {Ω0n }|n|≤K˜ ν lie in Λ1 , · · · , ΛLν , and, ν−1 Al 0 ν−1 ˜ ν−1 < |n| ≤ K ˜ν. ˜ , Ω0n ∈ Λl , K Al = 0 Ω0n 4.1. Iteration Lemma. The preceding analysis may be summarized as follows. Lemma 4.1. If ε is sufficiently small, then the following holds for all ν = 0, 1, · · · . a) Hν is real analytic on Dν × Oν , Lν+1
Nν
= eν + hων (ξ), Ii +
X l=1
Pν
= P˘ν + P´0 + P`0 ,
hA˜νl zlν+1 , z¯lν+1 i +
X ˜ ν+1 |n|>K
Ω0n wn w ¯n ,
,
20
JIANSHENG GENG AND YINGFEI YI
and moreover, |ων+1 − ων |Oν ≤ c0 εν , kA˜ν+1 − A˜νl kOν ≤ c0 εν , l kXP ν kDν ,Oν ≤ εν , P˘ν = P˘ ν (θ, I, z ν , z¯ν , ξ) +
X
P˘nν (θ, I, z ν , z¯ν , ξ)wnαn w ¯nβn ,
˜ ν ,αn +βn ≥1 |n|>K
X
P´0 =
0 αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m ,
n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
P`0 =
X
O(|wn |3 ),
n
with ˜
kP˘nν (θ, I, z ν , z¯ν , ξ)kDν ,Oν ≤ e−(|n|−Kν ) , 0 kP´nm (ξ)kD ,O ≤ e−|n−m| . ν
ν
b) There is a symplectic transformation ˜ ν × Oν+1 → Dν Φν : D such that Hν+1 = Hν ◦ Φν . Proof. It is sufficient to verify the conditions C1)–C6) for all ν = 0, 1, · · · , which are easily seen to follow from the following conditions D1) rν −r2 ν+1 ln 15 ≤ Kν+1 ≤ 11 , 4
εν4
εν4τ d
1 4
4
D2) c0 γ −d (rν − rν+1 )−N εν ≤ 12 for all ν = 0, 1, · · · . We first let ε (hence ε0 ) be sufficiently small such that 4
ε0 < min{ where Ψ(r0 ) =
δ γ 5d r05N 1 , }, 5 9N +6 2 c0 Ψ(r0 ) 2
∞ Y
4 i
[(ri−1 − ri )−5N ]( 5 )
i=1
which is easily seen to be well-defined. Then 1
4
c0 γ −d (r0 − r1 )−N ε04 ≤
1 , 2
i.e., D2) holds for ν = 0. Recall that 1 8 1 ln ≤ K0 ≤ . 1 r0 ε 54 ε04τ d4 0 We see that D1) also holds for ν = 0. Now, for any ν ≥ 1, we have by induction that 1
4
4
5
4
1
4 c0 γ −d (rν − rν+1 )−N εν4 = c0 γ −d (rν − rν+1 )−N (4c0 γ −d (rν−1 − rν )−N εν−1 )4 5
4
1
4
1
5 ν
4 ≤ (24N +2 c50 γ −5d (rν−1 − rν )−5N εν−1 ) 4 ≤ (24N +2 c50 γ −5d Ψ(r0 )ε0 ) 4 ( 4 )
≤ (
r05N 25N +4
1
5 ν
)4(4) ≤
1 , 2
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
21
and 2 2ν+3 1 1 3ν 1 ln 5 ≤ ln ( 5 )ν ≤ ln ≤ 4ν K 0 rν − rν+1 εν4 r0 r ε 4 0 0 ε0 1 1 , = Kν+1 ≤ ( 6 )ν 1 ≤ 1 ε0 5 4τ d4 εν4τ d4 i.e., D1) and D2) hold true.
ν
= Φ0 ◦ Φ1 ◦ · · · ◦ Φν−1 , ν = 1, 2, · · · . Inductively, we have that
4.2. Convergence. Let Ψ ˜ ν × Oν+1 → D0 and Ψν : D
H0 ◦ Ψν = Hν = Nν + P ν for all ν = 1, 2, · · · . ˜ = ∩∞ Oν . Applying Lemma 4.1 and standard arguments (e.g. [22, 27]) we conclude that Let O ν=0 ˜ say to, H∞ , e∞ , N∞ , P ∞ , Ψ∞ , Hν , eν , Nν , P ν , Ψν , DΨν , ων converge uniformly on D( 21 r0 , 0) × O, ∞ DΨ , ω∞ respectively. It is clear that ∞ X ∞ ∞ N∞ = e∞ + hω∞ , Ii + hA∞ ¯l i. l zl , z l=1
Since
5
4
4
5 ν
4 ≤ (4c0 γ −d Ψ(r0 )ε0 )( 4 ) , εν = 4c0 γ −d (rν−1 − rν )−N εν−1 we have by Lemma 4.1 that XP ∞ |D( 1 r0 ,0)×O˜ ≡ 0. 2
Let ΦtH denote the flow of any Hamiltonian vector field XH . Since H0 ◦ Ψν = Hν , we have that ΦtH0 ◦ Ψν = Ψν ◦ ΦtHν .
(4.1)
The uniform convergence of Ψν , DΨν , XHν imply that one can pass the limit in the above to conclude that ΦtH0 ◦ Ψ∞ = Ψ∞ ◦ ΦtH∞ , ˜ It follows that on D( 1 r0 , 0) × O. 2
ΦtH0 (Ψ∞ (TN × {ξ})) = Ψ∞ ΦtN∞ (TN × {ξ}) = Ψ∞ (TN × {ξ}), ˜ Hence Ψ∞ (TN × {ξ}) is an embedded invariant torus of the original perturbed for all ξ ∈ O. ˜ The frequencies ω∞ (ξ) associated with Ψ∞ (TN × {ξ}) are slightly Hamiltonian system at ξ ∈ O. deformed from the unperturbed ones ω(ξ), and, the normal behaviors of the invariant tori Ψ∞ (TN × {ξ}) are governed by their respective normal frequency matrices A∞ l . 4.3. Measure estimates. For each ν = 1, 2, · · · , let Kντ d2 ν −1 ) , Rk (γ) = ξ ∈ Oν−1 : |hk, ων−1 i | > ( γ Kντ d2 −1 ) , ) k > ( Rνkl (γ) = ξ ∈ Oν−1 : k(hk, ων−1 iI + A˜ν−1 l γ Kτ 2 ν+ Rklj (γ) = {ξ ∈ Oν−1 : k(hk, ων−1 iI + A˜ν−1 ⊗ I + I ⊗ A˜ν−1 )−1 k > ( ν )d }, j l γ and, τ ˜ν−1 ⊗ I − I ⊗ A˜ν−1 )−1 k > ( Kν )d2 }, |k| + |l − j| 6= 0. Rν− j klj (γ) = {ξ ∈ Oν−1 : k(hk, ων−1 iI + Al γ Then [ [ [ [ [ Oν ⊆ Oν−1 \(( Rνk (γ)) ( Rνkl (γ)) ( Rν± klj (γ))), |k|≤Kν
˜ν |k|≤Kν ,l≤K
˜ν |k|≤Kν ,l,j≤K
22
JIANSHENG GENG AND YINGFEI YI
for all ν = 1, 2, · · · . Consider the resonant sets [ [ [ Rνk (γ)) ( Rν = (
Rνkl (γ))
[
˜ν |k|≤Kν ,l≤K
|k|≤Kν
[
(
Rν± klj (γ)).
˜ν |k|≤Kν ,l,j≤K
It is clear that ˜⊆ O\O
[
Rν .
ν≥1
Lemma 4.2. There is a constant C1 > 0 such that [ [ meas(Rνk (γ) Rνkl (γ) Rν± klj (γ)) ≤ C1
γ Kντ −1
˜ ν , and ν = 1, 2, · · · . for all |k| ≤ Kν , l, j ≤ K Proof. The proof follows from arguments in [31]. For simplicity, we only estimate the measures of ν+ ν ν Rν− klj (γ). Measure estimates for Rk , Rkl , and Rklj (γ) can be obtained similarly. We note that (4.2) hk, ων (ξ)iI + A˜ν−1 ⊗I −I ⊗ A˜ν−1 = hk, ων−1 (ξ)iI +diag(Ωn −Ωm , · · · , Ωn −Ωm )+W (ξ), j
l
1
1
r
r
where Ωn1 , Ωm1 , · · · , Ωnr , Ωmr are unperturbed frequencies independent of parameters, and dim(A˜ν−1 ⊗ I) = r ≤ d2 , and kW (ξ)kOν−1 ≤ ε0 . l In the case k = 0 and l 6= j, since by assumption (A2) |Ωnj − Ωmj | > γ, j = 1, · · · , r, we have 2 Kτ by the standard Neumann series expansion that kA˜ν−1 ⊗ I − I ⊗ A˜ν−1 )−1 k < 2 < ( ν+1 )d as j
l
γ
γ
ε 1, i.e., Rν− 0lj (γ) = ∅. In the case k 6= 0, we have by Lemma 5.7 that ˜ν−1 ⊗ I − I ⊗ A˜ν−1 )| < ( Rν2 klj ⊂ {ξ ∈ Oν−1 : | det(hk, ων−1 (ξ)iI + Al j
γ d2 τ −1 ) }. Kν
Denote g(ξ) = det(hk, ων−1 (ξ)iI + A˜ν−1 ⊗ I − I ⊗ A˜ν−1 ). j l Then it follows from (4.2) that r Y X g(ξ) = (hk, ων−1 (ξ)i + Ωni − Ωmi ) + aα (hk, ων−1 (ξ)i + Ωn − Ωm )α , α
i=1
where kaα kOν−1 ≤ ε0 , the multi-index α runs over the set α = (α1 , · · · , αr ), αi = 0 or 1, and Pr α i=1 i ≤ r − 1. Due to the choice of ε0 , we have that |∂ξr g(ξ)|
≥
r Y
|hk, ∂ξ ων (ξ)i| − ε0 |k|r−1 ≥ (δ − ε0 )|k|r − ε0 |k|r−1 ≥
i=1
Hence by Lemma 5.8, meas(Rν− klj ) ≤ C1 [(
δ r |k| . 2
γ d2 1 γ ) ] r ≤ C1 τ −1 . Kντ −1 Kν
Lemma 4.3. ˜ ≤ meas( meas(O \ O)
[
Rν ) = O(γ).
ν≥1
Proof. By Lemma 4.2, there exists a constant C2 > 0 such that [ X Rν± C1 meas( klj ) ≤ ˜ν |k|≤Kν ,l,j≤K
˜ν |k|≤Kν ,l,j≤K
≤ 4C1
X |k|≤Kν
γ Kντ −1
γ Kντ −3
≤ C2
γ Kντ −b−3
.
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
Similarly, there are constants C3 > 0, C4 > 0 such that [ meas( Rνk (γ) ≤ C3 |k|≤Kν
γ
[
meas(
Rνkl (γ)) ≤ C4
˜ν |k|≤Kν ,l≤K
,
Kντ −b−3
and,
23
γ . Kντ −b−3
Let τ ≥ b + 4. We have that ˜ ≤ meas( meas(O \ O)
[
Rν )
ν≥1
=
meas[
[
((
ν≥1
= O(
ν≥0
Rνk )
[
Kν+1
[
(
Rνkl )
[
˜ν |k|≤Kν ,l≤K
|k|≤Kν
γ
X
[
(
[
Rν± klj ))]
˜ν |k|≤Kν ,l,j≤K
) = O(γ).
This completes the measure estimate. 5. Appendix Lemma 5.1. kF GkD(r,s),O ≤ kF kD(r,s),O kGkD(r,s),O . Proof. Since (F G)klαβ = kF GkD(r,s),O
=
P
sup kwk<s kwk<s ¯
≤
sup kwk<s kwk<s ¯
≤
Fk−k0 ,l−l0 ,α−α0 ,β−β 0 Gk0 l0 α0 β 0 , we have X |(F G)klαβ |s2l |wα ||w ¯ β |e|k|r
k0 ,l0 ,α0 ,β 0
k,l,α,β
X
X
k,l,α,β
k0 ,l0 ,α0 ,β 0
|Fk−k0 ,l−l0 ,α−α0 ,β−β 0 Gk0 l0 α0 β 0 |s2l |wα ||w ¯ β |e|k|r
kF kD(r,s),O kGkD(r,s),O .
Lemma 5.2. (Generalized Cauchy inequalities) 1 kF kD(r,s) , σ 4 kFI kD(r, 21 s) ≤ 2 kF kD(r,s) , s 2 kFw kD(r, 12 s) ≤ kF kD(r,s) , s 2 kFw¯ kD(r, 12 s) ≤ kF kD(r,s) . s
kFθ kD(r−σ,s) ≤
Proof. See [27].
Let {·, ·} denote the Poisson bracket of smooth functions: X ∂F ∂G ∂F ∂G ∂F ∂G ∂F ∂G {F, G} = h , i−h , i+i ( − ). ∂I ∂θ ∂θ ∂I ∂wn ∂ w ¯n ∂w ¯n ∂wn n Lemma 5.3. There exists a constant c > 0 such that if kFn kD(r,s) < e−|n| , kGkD(r,s) < ε, then k{Fn , G}kD(r−σ, 12 s) < cσ −1 s−2 kFn kD(r,s) kGkD(r,s) ≤ cσ −1 s−2 εe−|n| .
24
JIANSHENG GENG AND YINGFEI YI
Proof. By Lemmas 5.1, 5.2, khFnI , Gθ ikD(r−σ, 21 s) khFnθ , GI ikD(r−σ, 12 s) X k Fnwm Gw¯m kD(r, 21 s)
< 4σ −1 s−2 kFn k · kGk, cσ −1 s−2 kFn k · kGk, X ≤ kFnwm kD(r, 21 s) kGw¯m kD(r, 12 s)
0 such that if kXF kD(r,s) < ε0 , kXG kD(r,s) < ε00 for some ε0 , ε00 > 0, then kX{F,G} kD(r−σ,ηs) < cσ −1 η −2 ε0 ε00 , 3
1
for any 0 < σ < r and 0 < η 1. In particular, if η ∼ ε 4 , ε0 ∼ ε, ε00 ∼ ε 4 , then 5
kX{F,G} kD(r−σ,ηs) ∼ ε 4 . Proof. See [16].
The following lemmas can be found in the Appendix of [12, 32].
Lemma 5.5. Let O be a compact set in RN for which small divisor conditions hold. Suppose that 2 2 ω(ξ) are C d Whitney-smooth functions in ξ ∈ O with derivative bounded by L and f (ξ) are C d d2 norm bounded by L. Then Whitney-smooth functions in ξ ∈ O with CW g(ξ) ≡
f (ξ) hk, ω(ξ)i
2
is C d Whitney-smooth in O with 4
4
τd kgkO < cγ −d K+ L.
Lemma 5.6. Let O be a compact set in RN for which small divisor conditions hold. Suppose 2 2 that Al (ξ), Rl (ξ) are respectively C d Whitney-smooth matrices and vectors, and ω(ξ) is a C d Whitney-smooth function with derivatives bounded by L. Then Fl (ξ) = M −1 Rl (ξ) 2
is C d Whitney-smooth with 4
4
τd kFl kO ≤ cγ −d K+ L, where M stands for either hk, ωiI + A˜l or hk, ωiI ± A˜l ⊗ I ± I ⊗ A˜j .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
25
Lemma 5.7. Let M be a r × r non-singular matrix with kM k < |k|. Then {ξ : kM −1 k > h} ⊆ {ξ : | det M |
0. If for some constant d > 0, |g (m) (u)| ≥ d for all u ∈ I, then 1 meas(Ih ) ≤ ch m , where c = 2(2 + 3 + · · · + m + d−1 ). References [1] S. Aubry, Anti–integrability in dynamical and variational problems, Physica D 86 (1995), 284–296 [2] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D 103 (1997), 201–250 [3] D. Bambusi, Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators, Nonlinearity 9 (1996), 433–457 [4] D. Bambusi and D. Vella, Quasi periodic breathers in Hamiltonian lattices with symmetries, Discrete Cont. Dyn. Sys. Ser. B 2 (2002), 389–399 [5] J. Bourgain, Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schr¨ odinger equations, Annals of Math. 148 (1998), 363–439 [6] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (1995), 629–639 [7] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE., Int. Math. Res. Notices (1994), 475–497 [8] J. Bourgain, Green’s function estimates for lattice Schr¨ odinger operators and applications. Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005 [9] J. Bourgain, Nonlinear Schr¨ odinger equations, Park City Series 5, American Mathematical Society, Providence, Rhode Island, 1999 [10] O. M. Braun and Y. S. Kivshar, Nonlinear dynamics of the Frenkel-Kontorova model, Phys. Reports 306 (1998), 1-108 [11] A. Carpio, L. L. Bonilla, Oscillatory wave fronts in chains of coupled nonlinear oscillators, Phys. Rev. E 67 (2003), 656-621 [12] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys. 211 (2000), 498-525 [13] W. Craig and C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure. Appl. Math. 46 (1993), 1409-1498 [14] T. Dauxois, M. Peyrard and C. R. Willis, Localised breather-like solution in a discrete Klein-Gordon model and application to DNA, Physica D 57 (1992), 267-282 [15] J. Fr¨ ohlich, T. Spencer and C. E. Wayne, Localization in disordered nonlinear dynamical systems, J. Stat. Phys. 42 (1986), 247–274 [16] J. Geng and J. You, A KAM theorem for one dimensional Schr¨ odinger equation with periodic boundary conditions, J. Differentail Equations 209(2005), 1-56 [17] J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Commun. Math. Phys. 262(2006), 343-372 [18] J. Geng and Y. Yi, Quasi-periodic solutions in a nonlinear Schr¨ odinger equation, submitted [19] M. Johansson and S. Aubry, Existence and stability of quasiperiodic breathers in the discrete nonlinear Schr¨ odinger equation, Nonlinearity 10 (1997), 1151–1178 [20] O. Kresse and L. Truskinovsky, Mobility of lattice defects: discrete and continuum approaches, J. Mech. Phys. Solids, 51 (2003), 1305-1332 [21] S. B. Kuksin, Hamiltonian perturbations of infinite–dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl. 21 (1987), 192-205 [22] S. B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, 1556, Springer, Berlin, 1993 [23] S. B. Kuksin and J. P¨ oschel, Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schr¨ odinger equation, Annals of Math. 143 (1996), 149-179 [24] Y. Li and Y. Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, Trans. Amer. Math. Soc. 357 No. 4 (2004), 1565-1600 [25] R. S. Mackay and S. Aubry, Proof of existence of breathers of time–reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7 (1994), 1623–1643 [26] J. P¨ oschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici 71 (1996), 269296
26
JIANSHENG GENG AND YINGFEI YI
[27] J. P¨ oschel, A KAM Theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 119-148 [28] J. P¨ oschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Comm. Math. Phys. 127 (1990), 351–393 [29] E. W. Prohofsky, K. C. Lu, L. L. Van Zandt, and B. F. Putnam, Breathing models and induced resonant melting of the double helix, Phys. Lett. A 70 (1979), 492-494 [30] C. E. Wayne, Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory, Commun. Math. Phys. 127 (1990), 479-528 [31] J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov’s non-resonance condition, J. Math. Pures Appl. 80 (2001), 1045-1067 [32] J. You, Perturbation of lower dimensional tori for Hamiltonian systems, J. Differential Equations 152 (1999), 1-29 [33] X. Yuan, Construction of quasi–periodic breathers via KAM technique, Commun. Math. Phys. 226 (2002), 61–100 [34] C. T. Zhang, Soliton excitations in deoxyribonucleic acid (DNA) double holices, Phys. Rev. A 35 (1987), 886-891 J. Geng: Department of Mathematics, Nanjing University, Nanjing 210093, P.R.China E-mail address: [email protected] (J. Geng) Y. Yi: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 E-mail address: [email protected] (Y. Yi)
Abstract. We consider Hamiltonian networks of long ranged and weakly coupled oscillators with variable frequencies. By deriving an abstract infinite dimensional KAM type of theorem, we show that for any given positive integer N and a fixed, positive measure set O of N variable frequencies, there is a subset O∗ ⊂ O of positive measure such that each ω ∈ O∗ corresponds to a small amplitude, quasi-periodic breather (i.e., a solution which is quasi-periodic in time and exponentially localized in space) of the Hamiltonian network with N -frequencies which are slightly deformed from ω.
1. Introduction and Main Result P Associated with the symplectic structure n pn ∧ qn , we consider Hamiltonian networks defined by real analytic Hamiltonians of the form X p2 (1.1) H= ( n + Vn (qn )) + W ({qn }), 2 n∈Z
β2
where Vn ’s are the on-site potentials satisfying Vn (0) = Vn0 (0) = 0 and Vn00 (0) ≡ 2n , βn > 0, and W is a coupling potential. Hamiltonian networks have been used in solid state physics in describing the vibration of particles (atoms) in a lattice (see [10, 11, 20]) and also used to model DNA chains (see [10, 14, 34]). They also arise naturally as spatial discretization of Hamiltonian PDEs such as nonlinear wave equations. Among the solutions of a Hamiltonian network, of particular physical interests are the socalled breathers or quasi-periodic breathers, which are self-localized, time periodic or quasi-periodic, solutions whose amplitudes decay at least exponentially as |n| → ∞. Breathers or quasi-periodic breathers are often referred to as dynamical solitons or intrinsic localized modes in physics and they have been largely found via numerics in many physical models (see [10, 29] and references therein). The existence of breathers in Hamiltonian networks associated with Hamiltonians (1.1) was rigorously analyzed when βn ≡ β by Aubry [1, 2], Mackay–Aubry [25] for the inter-particle, nearest neighbor coupling potential X W ({qn }) = (qn+1 − qn )2 n
and by Bambusi [3] for the long–range coupling potential X 1 W ({qn }) = (qn − qm )2 , α > 1. |n − m|α n6=m
Like in [25], breathers in the nearest neighbor coupling case can be studied near a fixed periodic orbit of the uncoupled Hamiltonian by certain continuation or perturbation arguments, provided that the couplings are “weak”, and, no small divisors need to be considered in such perturbation problems. These perturbation techniques are also applicable in finding quasi-periodic breathers with 1991 Mathematics Subject Classification. Primary 37K60, 37K55. Key words and phrases. Coupled oscillators, Hamiltonian networks, long ranged coupling, KAM theory, quasiperiodic breathers. The second author is Partially supported by NSF grant DMS0204119. 1
2
JIANSHENG GENG AND YINGFEI YI
two or three frequencies for certain models with symmetries (see Bambusi–Vella [4], Johansson– Aubry [19]). Using a modified KAM technique, the existence of quasi-periodic breathers with any finite number of frequencies was recently shown by Yuan [33] for the higher order, nearest-neighbor coupling potential X (1.2) W ({qn }) = (qn+1 − qn )3 . n
Almost periodic breathers with infinitely many frequencies have also been investigated. Associated with the potential (1.2), Fr¨ ohlich-Spencer-Wayne [15] considered the case when the frequencies are non-negative random variables with smooth distribution of fast decay at infinity and showed that there is a set Ω ⊂ R∞ + with positive probability measure such that each ω ∈ Ω corresponds to an almost periodic breather with infinite many frequencies (see also P¨oschel [28] for more general spatial structures). In this paper, we will study the existence of quasi-periodic breathers for the Hamiltonian (1.1) with the following higher order, long-ranged coupling potential 1 X −|n−m|α e (qn − qm )3 , α ≥ 1, (1.3) W ({qn }) = 3 n6=m
or equivalently the Hamiltonian network X α d 2 qn + Vn0 (qn ) = − e−|n−m| (qn − qm )2 . 2 dt
(1.4)
m∈Z
For a given integer N > 1, we specify N integers {i1 , · · · , iN } and let Z1 = Z \ {i1 , · · · , iN }. N and assume the We treat ω = (βi1 , · · · , βiN ) as parameters in a bounded closed region O in R+ following spectral gap condition: SG) There exist 1 ≤ d < ∞ and γ > 0 such that {βn }n∈Z1 = ∪∞ l=1 Λl where Λl , l = 1, 2, · · · , are sets satisfying #(Λl ) ≤ d,
for all l,
and |βn − βm | ≥ γ,
for all βn ∈ Λl , βm ∈ Λj , l 6= j.
We will show the following result. Theorem A. Assume SG) with γ sufficiently small. Then there exists a Cantor set Oγ ⊂ O, with meas(O \ Oγ ) = O(γ), such that for any ω ∈ Oγ , the Hamiltonian network (1.4) associated with ω admits a small amplitude, linearly stable, quasi-periodic breather q(t) = ({qn (t)}) of N -frequency ω∗ which is close to ω, and moreover, |qn | ∼ e−|n| . The condition SG) clearly holds when βn = |n|, n ∈ Z1 . Comparing with the cases of nonlinear wave equations [12, 17, 26, 27] in which βn ∼ |n|, n ∈ Z, the validity of Theorem A crucially depends on the coupling potential or perturbation (1.3) which admits a weaker regularity but a higher order perturbation. For the case of nearest-neighboring coupled Hamiltonian networks, breathers were shown to be super-exponentially localized in space ([33]). This is due to the at most linear growth of the normal components in the normal form associated with the short-ranged coupling potential (1.2). Our result only asserts the exponential localization of quasi-periodic breathers due to the exponential growth of the normal components in the normal form associated with the exponentially weighted, long-ranged coupling potential (1.3). If the long-ranged coupling potential 1 1 X (qn − qm )3 , α > 1 W ({qn }) = 3 |n − m|α n6=m
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
3
is considered instead, then the normal components in the associated normal form will have a superexponential growth, and our method will equally applicable to yield quasi-periodic breathers which are localized like |n|1 α in space. Theorem A will be proved by using KAM (Kolmogorov-Arnold-Moser) method. In fact, we will present an abstract infinite dimensional KAM type of theorem from which Theorem A will follow. Such an infinite dimensional KAM theorem differs significantly from those for Hamiltonian PDEs like nonlinear Sch¨ odinger, wave, beam, and KdV equations studied by many authors using either KAM or CWB (Craig-Wayne-Bourgain) method (see [5, 6, 7, 8, 9, 12, 13, 16, 17, 18, 21, 22, 23, 26, 27, 30] and references therein). This is mainly due to the fact that, when the normal frequencies of a Hamiltonian network have linear growth, the perturbation (1.3) admits weaker regularity than those of Hamiltonian PDEs under KAM or Newton iterations. Similar to the short-ranged coupling cases considered in [25, 33], it is also important to study the existence of quasi-periodic breathers for Hamiltonian networks with long-ranged coupling potentials and constant frequencies βn ≡ β, n ∈ Z, i.e., those formed by weakly coupled identical oscillators. As the KAM iteration mechanism and measure estimates for the constant frequency case significantly differ from the variable ones to be studied in this paper, we will consider the constant frequency case in a separate work. The paper is organized as follows. In Section 2 we state an abstract infinite dimensional KAM theorem and prove Theorem A as a corollary. Sections 3 and 4 are devoted to the proof of the abstract KAM theorem. More precisely, in Section 3, we give detailed construction of the KAM iteration for one KAM step. We complete the proof of the abstract infinite dimensional KAM theorem in Section 4 by showing an iteration lemma, convergence, and measure estimate. Some technical lemmas are provided in the Appendix. 2. An Abstract KAM Theorem In this section, we will formulate an abstract KAM theorem which can be applied to the Hamiltonian networks of long-ranged and weakly coupled oscillators with variable frequencies. Theorem A will be proved by using the abstract KAM theorem and normal form reductions. 2.1. The abstract theorem. We begin with some notations. Let integers N > 1, d ≥ 1, and real numbers r, s > 0 be given. We consider the complex neighborhood D(r, s) of TN ×{0}×{0}×{0} ⊂ TN × RN × `1 × `1 defined by D(r, s) = {(θ, I, w, w) ¯ : |Imθ| < r, |I| < s2 , kwk < s, kwk ¯ < s}, where | · | denote the sup-norm of complex vectors and k · k denote the `1 norm. Also let O be a positive (Lebesgue) measure set in RN . Let F (θ, I, w, w) ¯ be a real analytic function on D(r, s) which depends on a parameter ξ ∈ O, 2 2 C d -Whitney smoothly (i.e., C d in the sense of Whitney). We expand F into the Taylor-Fourier series with respect to θ, I, w, w: ¯ X F (θ, I, w, w) ¯ = Fαβ wα w ¯β , α,β
where α ≡ (· · · , αn , · · · ), β ≡ (· · · , βn , · · · ), αn , βn ∈ N, are multi-indices with finitely many non-vanishing components, and X Fαβ = Fklαβ (ξ)I l eihk,θi . k∈ZN ,l∈NN
We define the weighted norm of F by kF kD(r,s),O ≡ sup kwk<s kwk<s ¯
X α,β
kFαβ k |wα ||w ¯ β |,
4
JIANSHENG GENG AND YINGFEI YI
where kFαβ k ≡
X
|Fklαβ |O s2|l| e|k|r ,
|Fklαβ |O ≡ sup { max2 |∂ξm Fklαβ |}. ξ∈O m≤d
k,l
In the above and also for the rest of the paper, derivatives in ξ ∈ O are taken in the sense of Whitney. For a vector-valued function G : D(r, s) × O → Cn , n < ∞, we simply define its weighed norm by n X kGkD(r,s),O ≡ kGi kD(r,s),O . i=1
For the Hamiltonian vector field XF = (FI , −Fθ , {iFwn }, {−iFw¯n }) associated with a Hamiltonian function F on D(r, s) × O, we define its weighted norm by X 1 X 1 kFwn kD(r,s),O + kFw¯n kD(r,s),O ). kXF kD(r,s),O ≡ kFI kD(r,s),O + 2 kFθ kD(r,s),O + ( s s n n X Associated with the symplectic structure dI ∧ dθ + i dwn ∧ dw ¯n , we consider the following n∈Z
family of real analytic, parameterized Hamiltonians (2.1)
H
= N + P,
N
= hω(ξ), Ii +
X
Ωn wn w ¯n ,
n∈Z
P
= P (θ, I, w, w, ¯ ξ),
where (I, θ, w, w) ¯ ∈ D(r, s), ξ ∈ O, Ωn ’s are positive and independent of ξ, and all ξ-dependence 2 are of class C d in the sense of Whitney. It is clear that when P = 0, the unperturbed Hamiltonians N are completely integrable, admitting a family of quasi-periodic solutions (θ + ωt, 0, 0, 0) corresponding to invariant N -tori in the phase space. To study the persistence of some of these N -tori, we need the following assumptions on ω(ξ), Ωn and the perturbation P : (A1) Non-degeneracy of tangential frequencies: There is a constant δ > 0 such that ∂ω | det ( )| ≥ δ. ∂ξ (A2) Gap conditions of normal frequencies: There exist sufficiently small γ > 0 and sets Λl , l = 1, 2, · · · , such that {Ωn }n∈Z = ∪∞ l=1 Λl , #(Λl ) ≤ d, for all l, |Ωn − Ωm | ≥ γ, if Ωn ∈ Λl , Ωm ∈ Λj , l 6= j.
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
5
(A3) Decay property of the perturbation: P = P˘ + P´ + P` , where P˘ = P˘ (θ, I, w, w, ¯ ξ), P´ = ´ ` ` P (θ, I, w, w, ¯ ξ), P = P (θ, I, w, w, ¯ ξ) are such that X (2.2) P˘ = P˘ (θ, I, 0, 0, ξ) + P˘n (θ, I, ξ)wnαn w ¯nβn , kP˘n (θ, I, ξ)k ≤ e−|n| ; n∈Z αn +βn ≥1
(2.3)
P´
X
=
αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m ,
kP´nm (ξ)k ≤ e−|n−m| ;
n,m∈Z,n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
(2.4)
P`
=
X
O(|wn |3 ).
n∈Z
Our abstract KAM theorem states as the following. Theorem B. Consider the Hamiltonian (2.1) and assume (A1)-(A3). For a fixed γ > 0 sufficiently small, there exists a positive constant ε = ε(O, d, δ, N, γ, r, s) such that if kXP kD(r,s),O < ε, then the following holds. There exist Cantor sets Oγ ⊂ O with meas(O \ Oγ ) = O(γ) and maps Ψ : TN × Oγ → D(r, s),
ω ˜ : Oγ → R N ,
2
which are real analytic in θ and C d -Whitney smooth in ξ with kΨ − Ψ0 kD( r2 ,0),Oγ → 0 and |˜ ω − ω|Oγ → 0 as γ → 0, where Ψ0 is the trivial embedding: TN × O → TN × {0, 0, 0}, such that each ξ ∈ Oγ and θ ∈ TN corresponds to a linearly stable, N -frequency quasi-periodic solution Ψ(θ + ω ˜ (ξ)t, ξ) = (θ + ω ˜ (ξ)t, I(t), {wn (t)}) of the Hamiltonian (2.1). Moreover, |wn | ∼ e−|n| . Since our perturbation has a weaker regularity, the frequencies of these invariant tori are in general non-resonant instead of Diophantine. Comparing with results on quasi-periodic solutions for Hamiltonian PDEs (see e.g. [5, 6, 7, 8, 9, 12, 13, 16, 17, 18, 21, 22, 23, 26, 27, 30]), the above theorem relaxes the linear or super-linear growth conditions on the normal frequencies Ωn . Indeed, it is easy to see that the gap condition (A2) above is weaker than the linear or sup-linear growth conditions on the normal frequencies. The assumption (A3) is new but natural for networks of long ranged and weakly coupled oscillators. It is not clear whether a Lyapunov center theorem is possible for a Hamiltonian network whose normal frequencies satisfy the gap condition (A2). At least, the above theorem assert a quasiperiodic type of Lyapunov center result in the sense of measure. 2.2. Proof of Theorem A. Recall that the Hamiltonian networks of long ranged, weakly coupled oscillators considered in Theorem A is described by the Hamiltonian X p2 1 X −|n−m|α H= [ n + Vn (qn )] + e (qn − qm )3 , α ≥ 1, 2 3 n∈Z
n6=m
which, in terms of the Taylor expansion at q = 0, can be equivalently rewritten as X p2 X β 2 q2 1 X −|n−m|α H= [ n + n n] + e (qn − qm )3 + O(|qn |3 ). 2 2 3 n∈Z
n∈Z
n6=m
Let ε > 0 be sufficiently small. With the re-scalings pn , qn → εpn , εqn , the re-scaled Hamiltonian reads X p2 X β 2 q2 ε X −|n−m|α ε−2 H(εp, εq) = [ n + n n] + e (qn − qm )3 + ε O(|qn |3 ). 2 2 3 n∈Z
n6=m
n∈Z
Let N , {i1 , · · · , iN }, and Z1 = Z \ {i1 , · · · , iN } be as in Theorem A. For a given value a = (a1 , · · · , aN ) ∈ RN ¯ = + , we introduce the standard action-angle-normal variables (I, θ, w, w)
6
JIANSHENG GENG AND YINGFEI YI
(I, θ, {wn }n∈Z1 , {w ¯n }n∈Z1 ) ∈ RN × TN × `1 , i.e., s q p 1 p pij = βij Ij + aj cos θj , qij = Ij + aj sin θj , 1 ≤ j ≤ N, βij √ βn (wn + w ¯n ) wn − w ¯n √ , n ∈ Z1 . pn = , qn = √ i 2βn 2 Let ξ = (ξ1 , · · · , ξN ) = (βi1 , · · · , βiN ). Then in terms of the action-angle-normal variables the above Hamiltonian becomes X Ωn wn w ¯n + P (θ, I, w, w, ¯ ξ), (2.5) H = N + P = hω(ξ), Ii + n∈Z1
where ω(ξ) = (ω1 (ξ), · · · , ωN (ξ)) = (ξ1 , · · · , ξN ), Ωn = βn , n ∈ Z1 , and P = P˘ + P´ + P` satisfying P˘
= P˘ (θ, I, 0, 0, ξ) +
X
P˘n (θ, I, ξ)wnαn w ¯nβn ,
α
kP˘n (θ, I, ξ)k ≤ e−|n| ≤ e−|n| ;
n∈Z1 αn +βn ≥1
P´
X
=
αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m ,
α
kP´nm (ξ)k ≤ e−|n−m| ≤ e−|n−m| ;
n,m∈Z1 ,n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
P`
=
X
O(|wn |3 ).
n∈Z1
It is also easy to see that we can choose appropriate r, s > 0 such that kXP kD(r,s),O < ε. Hence the Hamiltonian (2.5) satisfies all conditions of Theorem B, from which Theorem A follows. 3. KAM Step In what follows, we will perform KAM iterations to (2.1) which involves infinite many successive steps, called KAM steps, of iterations, to eliminate lower order θ-dependent terms in P . Each KAM step will make the perturbation smaller than the previous one at a cost of excluding a small measure set of parameters. At the end, the KAM iterations will be shown to converge and the measure of the total excluding set will be shown to be small. To begin with the KAM iteration, we set Ω0n = Ωn , n ∈ Z1 , r0 = r, s0 = s. 3.1. Normal form. We first convert the Hamiltonian (2.1) into a more convenient form in order 5 to perform the KAM iteration. Let ε∗ ∼ ε 4 and choose a K0 such that K0 ∼ | ln ε∗ |. According to the forms of (2.3), (2.4) in the Assumption (A3), we can make s0 smaller if necessary such that kXP´ +P` kD(r0 ,s0 )×O ≤ ε∗ . We now treat the term P˘ . According to the form of (2.2) and the definition of the norm, we have X P˘ = P˘ (θ, I, 0, 0, ξ) + P˘n (θ, I, ξ)wnαn w ¯nβn αn +βn ≥1
=
X
P˘kl I l eihk,θi +
k,l
X
P˘nklαn βn I l eihk,θi wnαn w ¯nβn ,
n,k,l αn +βn ≥1
where kP˘kl k ≤ e−|k|r0 ,
kP˘nklαn βn k ≤ e−|k|r0 e−|n| .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
7
Thus we can make r0 , s0 smaller such that kXP
|k|>K0 ,|l|≤1
P P˘kl I l eihk,θi + |n|>K0
or |k|>K0 αn +βn ≥1
αn βn P˘nkαn βn eihk,θi wn w ¯n +O(|I|2 )+O(|w|3 ) k
≤ ε∗ .
Let R=
X
X
P˘kl I l eihk,θi +
P˘nkαn βn eihk,θi wnαn w ¯nβn .
|n|≤K0 ,|k|≤K0 1≤αn +βn ≤2
|k|≤K0 ,|l|≤1
We first construct a symplectic transformation Φ∗ = Φ1F∗ defined as the time-1 map of the Hamiltonian flow associated to a Hamiltonian F∗ of the form X X F∗ = Fkl00 I l eihk,θi + (Fnk10 wn + Fnk01 w ¯n )eihk,θi |k|≤K0 ,|n|≤K0
0K0 αn +βn ≥1
satisfies kP˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)k ≤ e−(|n|−K0 ) . In the above, the first and the second term of P˘∗ come from P ◦ Φ∗ and P˘ ◦ Φ∗ + P´ ◦ Φ∗ respectively. The decay property of P˘n∗ follows from the fact that Φ∗ depends only on I, θ and wm , w ¯m for |m| ≤ K0 .
8
JIANSHENG GENG AND YINGFEI YI
Next, write X
P˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)wnαn w ¯nβn
|n|>K0 αn +βn ≥1
=
X
P˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)wnαn w ¯nβn
|n|>5K0 αn +βn ≥1
+
X
¯nβn . P˘n∗ (θ, I, wm(|m|≤K0 ) , w ¯m(|m|≤K0 ) , ξ)wnαn w
K0 K αn +βn ≥1
= P˘ ν (θ, I, z ν , z¯ν , ξ) +
X
P˘nν (θ, I, z ν , z¯ν , ξ)wnαn w ¯nβn
˜ν |n|>K αn +βn ≥1
˜ =K ˜ ν is a positive constant, L = Lν defined on a domain D(r, s) × O = D(rν , sν ) × Oν , where K 0 is the smallest positive integer such that {Ωn }|n|≤K˜ lie in Λ1 , · · · , ΛL , zl = zlν = (· · · , wn , · · · ) Ωn ∈Λl , ˜ |n|≤K
ν
z = z = (z1 , · · · , zL ), P = P ν , for some ε = εν , and
z¯l = z¯lν = (· · · , w ¯n , · · · ) Ωn ∈Λl , ˜ |n|≤K
ν
z¯ = z¯ = (¯ z1 , · · · , z¯L ), ˜
kP˘n (θ, I, z, z¯, ξ)kD(r,s),O ≤ e−(|n|−K) ,
˜ |n| > K.
˜ It is clear that L ≤ K. We will construct a symplectic transformation Φ = Φν , which, in smaller frequency and phase domains, carries the above Hamiltonian into the next KAM cycle. Below, all constants c1 − c12
10
JIANSHENG GENG AND YINGFEI YI
are positive and independent of the iteration process. The tensor product (or direct product) of two m × n, k × l matrices A = (aij ), B is a (mk) × (nl) matrix defined by a11 B · · · a1n B ··· ··· . A ⊗ B = (aij B) = · · · am1 B · · · amn B We also use k · k to denote the operator matrix norm, i.e., for a matrix M , kM k = supkyk=1 kM yk. ˜ < |n| ≤ K ˜ + will be added to the new ˜+ = 7 K ˜ + 1 K0 . In this KAM step, Ω0n with K Let K 4 4 normal matrix A+ according to the assumption (A2). In order to have a compact formulation l when solving a homological equation, we rewrite N as N
e + hω(ξ), Ii +
=
L X
hAl zl , z¯l i
l=1
X
+
X
Ω0n wn w ¯n +
Ω0n wn w ¯n
˜+ |n|>K
˜ ˜+ KK
l=1
where dim(A˜l ) ≤ d, L+ is the smallest positive integer such that {Ωn }|n|≤K˜ + lie in Λ1 , · · · , ΛL+ ˜ + ), (hence L+ ≤ K Al 0 ˜ < |n| ≤ K ˜ +, A˜l = , Ω0n ∈ Λl , K 0 Ω0n zl+ = (· · · , wn , · · · )
Ωn ∈Λl ˜ |n|≤K +
,
+ z + = (z1+ , · · · , zL ), +
z¯l+ = (· · · , w ¯n , · · · )
Ωn ∈Λl ˜ |n|≤K +
,
+ z¯+ = (¯ z1+ , · · · , z¯L ). +
3.2. Truncation. We first expand P˘ into the Taylor-Fourier series X X ¯nβn , P˘ = P˘klαβ eihk,θi I l z α z¯β + P˘klnαβ eihk,θi I l z α z¯β wnαn w k,l,α,β
k,l,n,α,β ˜ |n|>K,α n +βn ≥1
where k ∈ ZN , l ∈ NN and the multi-index α (resp. β) runs over the set α ≡ (α1 , · · · , αl , · · · , αL ) for αl = (· · · , αm , · · · ) Ωm ∈Λl , αm ∈ N (resp. β ≡ (β 1 , · · · , β l , · · · , β L ) for β l = (· · · , βm , · · · ) Ωm ∈Λl , ˜ |m|≤K
˜ |m|≤K
βm ∈ N). Let R be the following truncation of P˘ : X
R(θ, I, z, z¯, w, w) ¯ =
P˘kl00 eihk,θi I l
|k|≤K+ ,|l|≤1
+ +
X
(hP˘lk10 , zl i |k|≤K+ ,l≤L X
+ hP˘lk01 , z¯l i)eihk,θi +
X
(P˘nk10 wn + P˘nk01 w ¯n )eihk,θi
˜ ˜+ |k|≤K+ ,KK 1
where h.o.t. denotes the terms of the form O(|I|2 + |I||w| + |w|3 ). Let r+ = 2r + r40 , η = ε 4 . Using the facts ˜ kP˘n (θ, I, z, z¯, ξ)k ≤ e−(|n|−K) , kP˘k (I, z, z¯, ξ)k ≤ e−|k|r , we have that if r−r+ 5 C1) e−K+ 2 ≤ ε 4 ,
12
JIANSHENG GENG AND YINGFEI YI
then (3.3)
kXP −R kD(r
r−r+ ++ 2
,ηs),O
X
≤
e−|k|
r−r+ 2
˜
X
+
5
e−(|n|−K) + h.o.t. ≤ c1 ε 4 .
˜+ |n|>K
|k|>K+
3.3. The homological equation. We now look for a Hamiltonian F , defined in a domain D+ = D(r+ , s+ ), such that the time-1 map Φ = Φ1F of the Hamiltonian vector field XF defines a map from D+ to D and transforms H into H+ in the next KAM cycle. Let F have the form F (θ, I, z + , z¯+ )
= F0 + F1 + F2 X = Fkl00 eihk,θi I l +
(fnk10 wn + fnk01 w ¯n )eihk,θi
˜+ |k|≤K+ ,|n|≤K
0 . Similarly, Fljk11 = (Fjlk11 )> and Fljk02 = (Fjlk02 )> . Hence, the Hamiltonian F is uniquely determined on O+ . We proceed to estimate XF and Φ1F . . Lemma 3.3. Let Di = D(r+ + 4i (r − r+ ), 4i s), 0 < i ≤ 4. If 4
1
τd C2) K+ ≤ ε− 4 , then there is a constant c2 > 0 such that 4
3
kXF kD3 ,O+ ≤ c2 γ −d (r − r+ )−N ε 4 . Proof. By the definition of O+ , Lemma 3.1, Lemma 3.2 and Lemma 5.5, Lemma 5.6 in the Appendix, we have 2 4 τ d4 ˘ |Fkl00 |O+ ≤ |hk, ωi|−d |P˘kl00 |O+ ≤ γ −d K+ |Pkl00 |O+ , k 6= 0; 4
4
4
4
4
4
4
4
4
4
kFlk10 kO+
τd ≤ γ −d K+ kRlk10 kO+ ;
kFlk01 kO+
τd ≤ γ −d K+ kRlk01 kO+ ;
kFjlk20 kO+
τd k20 ≤ γ −d K+ kRjl kO+ ;
kFjlk11 kO+
τd k11 ≤ γ −d K+ kRjl kO+ , |k| + |l − j| = 6 0;
kFjlk02 kO+
τd k02 ≤ γ −d K+ kRjl kO+ .
It follows that 1 kFθ kD3 ,O+ s2
1 1 X ( |Fkl00 | · s2|l| · |k| · e|k|(r− 4 (r−r+ )) 2 s k,|l|≤1 X 1 + (kFlk10 k · kzl+ k) · |k| · e|k|(r− 4 (r−r+ ))
≤
k,l
+
X
+
X
1
(kFlk01 k · k¯ zl+ k) · |k| · e|k|(r− 4 (r−r+ ))
k,l 1
kFjlk20 k · kzl+ kkzj+ k · |k| · e|k|(r− 4 (r−r+ ))
k,l,j
+
X
1
kFjlk11 k · kzl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ ))
|k|+|l−j|6=0
+
X k,l,j
1
kFjlk02 k · k¯ zl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ )) )
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
15
4 τ d4 X γ −d K+ 1 ( |P˘kl00 | · s2|l| · |k| · e|k|(r− 4 (r−r+ )) 2 s k,|l|≤1 X 1 + (kRlk10 k · kzl+ k) · |k| · e|k|(r− 4 (r−r+ ))
≤
k,l
+
X
+
X
1
(kRlk01 k · k¯ zl+ k) · |k| · e|k|(r− 4 (r−r+ ))
k,l 1
k20 kRjl k · kzl+ kkzj+ k · |k| · e|k|(r− 4 (r−r+ ))
k,l,j 1
X
+
k11 kRjl k · kzl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ ))
|k|+|l−j|6=0
+
X
1
k02 kRjl k · k¯ zl+ kk¯ zj+ k · |k| · e|k|(r− 4 (r−r+ )) )
k,l,j 4
4
τd ≤ c3 γ −d (r − r+ )−N K+ kXR k 4
3
≤ c3 γ −d (r − r+ )−N ε 4 . Similarly, kFI kD3 ,O+
=
X
4
1
3
|Fkl00 |e|k|(r− 4 (r−r+ )) ≤ c4 γ −d (r − r+ )−N ε 4 .
|l|=1
kXF1 kD3 ,O+
≤
X X 1 X 1 X ( kF1wn k + kF1w¯n k) ≤ ( kF1z+ k + kF1z¯+ k) s n s l l n l
kXF2 kD3 ,O+
−d4
−N
τ d4 K+ kXR1 k
−d4
−N
τ d4 K+ kXR2 k
l
−d4
−N
−d4
−N
3
≤ c5 γ (r − r+ ) ≤ c5 γ (r − r+ ) ε 4 . X X 1 X 1 X ( kF2z+ k + kF2z¯+ k) ≤ kF2wn k + kF2w¯n k) ≤ ( s n s l l n l
≤ c6 γ
(r − r+ )
≤ c6 γ
l
(r − r+ )
3
ε4 .
The proof is now completed by adding the estimates above together. Let Diη = D(r+ + 4i (r − r+ ), 4i ηs), 0 < i ≤ 4.
Lemma 3.4. If 4 1 C3) c2 γ −d (r − r+ )−N ε 2 < 1, then ΦtF : D2η → D3η ,
(3.9)
−1 ≤ t ≤ 1,
and moreover, 4
3
kDΦtF − IkD1η < c7 γ −d (r − r+ )−N ε 4 .
(3.10) Proof. Let
kDm F kD,O+ = max{k
∂ |i|+|l|+|α|+|β| ∂θi ∂I l ∂(z + )α ∂(¯ z + )β
F kD,O+ , |i| + |l| + |α| + |β| = m ≥ 2}.
We note that F is a polynomial of order 1 in I and of order 2 in z + , z¯+ . It follows from Lemma 3.3 and the Cauchy inequality that 4
3
kDm F kD2 ,O+ < c8 γ −d (r − r+ )−N ε 4 , for any m ≥ 2. Using the integral equation ΦtF
Z = id + 0
t
XF ◦ ΦsF ds
16
JIANSHENG GENG AND YINGFEI YI
and Lemma 3.3, we easily see that ΦtF : D2η → D3η , −1 ≤ t ≤ 1. Since Z t Z t DΦtF = Id + (DXF )DΦsF ds = Id + J(D2 F )DΦsF ds, 0
0
where J denotes the standard symplectic matrix. Let c7 = 2c8 . It follows that 4
3
kDΦtF − Ik ≤ 2kD2 F k ≤ c7 γ −d (r − r+ )−N ε 4 . 3.4. The new Hamiltonian. Let Φ = Φ1F , s+ = 18 ηs, D+ = D(r+ , s+ ), and = e+ + hω+ , Ii +
N+
L+ X
X
+ + hA+ ¯l i + l zl , z
Ω0n wn w ¯n ,
˜+ |n|>K
l=1
= P˘+ + P´0 + P`0 ,
P+ where e+ A+ l zl+ +
z
P˘+
= e + P˘0000 , ω+ = ω + P˘0l00 (|l| = 1), = A˜l + R011 , l ≤ L+ , ll
=
(· · · , wn , · · · )|n|≤K˜ + ,
z¯l+ = (· · · , w ¯n , · · · )|n|≤K˜ + ,
+ + (z1+ , · · · , zL ), z¯+ = (¯ z1+ , · · · , z¯L ), + + Z 1 Z 1 Z 1 = (1 − t){{N, F }, F } ◦ ΦtF dt + {R, F } ◦ ΦtF dt + (P˘ − R) ◦ Φ1F + {P´0 + P`0 , F } ◦ ΦtF dt.
=
0
0
0
Then Φ : D+ × O+ → D, and, by the second order Taylor formula, H+
≡ H ◦ Φ = (N + R) ◦ Φ + (P − R) ◦ Φ Z 1 = N + {N, F } + R + (1 − t){{N, F }, F } ◦ ΦtF dt 0 1
Z
{R, F } ◦ ΦtF dt + (P˘ − R) ◦ Φ1F + (P´0 + P`0 ) ◦ Φ1F
+ 0
= N + {N, F } + R + P˘+ + P´0 + P`0 = N+ + P + + {N, F } + R − P˘0000 − hω 0 , Ii −
L+ X
011 + + hRll zl , z¯l i
l=1 +
= N+ + P . Below, we show that the new Hamiltonian H+ enjoys similar properties as H. By the Assumptions of P˘ , we have that there is a constant c9 > 0 such that |ω+ − ω|O ≤ c9 ε, kA+ − A˜l kO ≤ c9 ε. +
l
+
Denote R(t) = (1 − t)(N+ − N ) + tR. We can rewrite P + as Z 1 Z 1 + t P = (1 − t){{N, F }, F } ◦ ΦF dt + {R, F } ◦ ΦtF dt + (P − R) ◦ Φ1F 0
Z =
0 1
{R(t), F } ◦ ΦtF dt + (P − R) ◦ Φ1F .
0
Hence Z XP + = 0
By Lemma 3.4, if 4 3 C4) c7 γ −d (r − r+ )−N ε 4 ≤ 1,
1
(ΦtF )∗ X{R(t),F } dt + (Φ1F )∗ X(P −R) .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
17
then kDΦtF kD1η ≤ 1 + kDΦtF − IkD1η ≤ 2, By Lemma 5.4 and (3.3), we also have
−1 ≤ t ≤ 1.
4
7
kX{R(t),F } kD2η ≤ c10 γ −d (r − r+ )−N η −2 ε 4 , 5
kX(P −R) kD2η ≤ c1 ε 4 . Let c0 = max{c1 , · · · , c10 , c11 , c12 }, where c11 , c12 will be defined later, and let 4
5
ε+ = 4c0 γ −d (r − r+ )−N ε 4 . Then
4
5
5
kXP + kD+ ,O+ ≤ 2c1 ε 4 + 2c10 γ −d (r − r+ )−N ε 4 ≤ ε+ . We now exam the decay property of P˘+ . More precisely, write X P˘+ = P˘ + (θ, I, z + , z¯+ , ξ) + P˘n+ (θ, I, z + , z¯+ , ξ)wnαn w ¯nβn . ˜ + ,αn +βn ≥1 |n|>K
We show that
˜
kP˘n+ (θ, I, z + , z¯+ , ξ)kD+ ,O+ ≤ e−(|n|−K+ ) ,
|n| >˜˜K+ . ˜ + , so does {N, F }. Hence Since F only involves the normal components wn , w ¯n for |n| ≤ K R1 t ˜ + . Recall ¯n for |n| ≤ K (1 − t){{N, F }, F } ◦ ΦF dt only involves the normal components wn , w 0 that X P`0 = O(|wn |3 ). n
R ˜ + , so does 1 {P`0 , F }◦Φt dt. Hence {P`0 , F } only involves the normal components wn , w ¯n for |n| ≤ K F R1 0 Since R is a truncation of P˘ , we only need to consider the terms (P˘ − R) and 0 {P˘ + P´0 , F } ◦ ΦtF dt. Recall that X ¯nβn , P˘n (θ, I, z, z¯, ξ)wnαn w P˘ = P˘ (θ, I, z, z¯, ξ) + ˜ n +βn ≥1 |n|>K,α ˜
kP˘n (θ, I, z, z¯, ξ)kD(r,s),O+ ≤ e−(|n|−K) . X 0 αm βm w ¯m , P´0 = P´nm (ξ)wnαn w ¯nβn wm n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
0 kP´nm (ξ)kD(r,s),O+ ≤ e−|n−m| . ˜ + , the terms corresponding to the Since R only involves the normal components wn , w ¯n for |n| ≤ K ˜ ˘ ˜ + in P˘ , normal components wn , w ¯n for |n| > K+ in P − R are just those corresponding for |n| > K for which we have the decay property ˜
˜
kP˘n (θ, I, z, z¯, ξ)kD(r+ ,s+ ),O+ ≤ e−(|n|−K) ≤ e−(|n|−K+ ) . R1 To prove the decay estimates of 0 {P˘ + P´0 , F } ◦ ΦtF dt, we only need to consider the terms cor˜ + . Since F is independent of normal responding to the normal components wn , w ¯n for |n| > K R1 ˜ ˘ components wn , w ¯n for |n| > K+ , so is 0 {P (θ, I, z, z¯, ξ), F } ◦ ΦtF dt. Similarly, Z 1 X 0 αm βm { P´nm (ξ)wnαn w ¯nβn wm w ¯m , F } ◦ ΦtF dt 0
˜ n6=m,|n|,|m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
18
JIANSHENG GENG AND YINGFEI YI
˜ + . Thus, it remains to consider terms is independent of the normal components wn , w ¯n for |n| > K Z 1 X { P˘n (θ, I, z, z¯, ξ)wnαn w ¯nβn , F } ◦ ΦtF dt 0
˜ n +βn ≥1 |n|>K,α 1
Z
X
= 0
(3.11)
{P˘n (θ, I, z, z¯, ξ), F } ◦ ΦtF wnαn w ¯nβn dt
˜ n +βn ≥1 |n|>K,α
Z 1 ( {P˘n (θ, I, z, z¯, ξ), F } ◦ ΦtF dt)wnαn w ¯nβn ,
X
=
0
˜ n +βn ≥1 |n|>K,α
and 1
Z
X
{ 0
1
Z
X
= 0
(3.12)
˜+ |n|>K
X
{
˜+ |n|>K
X
=
0 αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m , F } ◦ ΦtF dt
˜ ,|m|≤K ˜ n6=m,|n|>K + + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
˜ |m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
Z 1 ( { 0
0 αm βm P´nm (ξ)wm w ¯m , F } ◦ ΦtF wnαn w ¯nβn dt
X
0 αm βm ¯nβn . w ¯m , F } ◦ ΦtF dt)wnαn w P´nm (ξ)wm
˜ |m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
Let P˜n = P˘n (θ, I, z + , z¯+ , ξ) +
X
0 P´nm (ξ).
˜ |m|≤K + αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
We combine (3.11) and (3.12) to consider decay property of Z 1 X ( {P˜n , F } ◦ ΦtF dt)wnαn w ¯nβn . ˜ n +βn ≥1 |n|>K,α
0
˜
˜
By relaxing decay properties of e−(|n|−K) , e−|n−m| to e−(|n|−K+ ) , we have by Lemma 5.3 that 4
3
˜
k{P˜n , F }kD(r−σ, 12 s) ≤ c11 γ −d (r − r+ )−N σ −1 s−2 ε 4 e−(|n|−K+ ) . It follows from Cauchy estimate that 4
3
˜
kX{P˜n ,F } kD(r−2σ, 14 s) ≤ c12 γ −d (r − r+ )−N σ −2 s−4 ε 4 e−(|n|−K+ ) . Hence by Lemma 3.4, if 4 3 C5) c11 γ −d (r − r+ )−N η −2 ε 4 ≤ 21 , 4
3
C6) c12 c2 (γ −d (r − r+ )−N η −2 ε 4 )2 ≤ 12 , then Z 1 k {P˜n , F } ◦ ΦtF dtkD(r+ ,s+ ) 0
k{P˜n , F } ◦ ΦtF kD(r+ ,s+ ) ≤ k{P˜n , F }kD(r+ ,s+ ) + k{P˜n , F } ◦ ΦtF − {P˜n , F }kD(r+ ,s+ ) ≤ k{P˜n , F }kD(r+ ,s+ ) + kX{P˜n ,F } kD2η kΦtF − idkD1η ≤
4
3
˜
4
3
˜
≤ c11 γ −d (r − r+ )−N η −2 ε 4 e−(|n|−K+ ) + c12 c2 (γ −d (r − r+ )−N η −2 ε 4 )2 e−(|n|−K+ ) ˜
≤ e−(|n|−K+ ) .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
This completes one step of KAM iterations.
19
4. Proof of Theorem B ˜ 0 , O0 , H0 be given at the beginning of Section 3. For each ν = 0, 1, · · · , we Let r0 , s0 , ε0 , γ, K0 , K label all index-free quantities by ν and label all +-indexed quantities by ν + 1. This defines, for all ν = 1, 2, · · · , the following sequences: rν
= r0 (1 −
ν+1 X
2−i ),
i=2 −d4
5
4 , (rν−1 − rν )−N εν−1
εν
=
4c0 γ
sν
=
ν−1 Y 1 1 1 ην−1 sν−1 = 2−3ν ( εi ) 4 s0 , ην = εν4 , 8 i=0
Kν ˜ν K
=
Dν ˜ν D Hν Nν
4Kν−1 , ˜ ν−1 + Kν , = K = D(rν , sν ), 1 1 = D(rν+1 + (rν − rν+1 ), ην sν ), 4 4 = Nν + Pν , Lν X X hAνl zlν , z¯lν i + = eν + hων (ξ), Ii + Ω0n wn w ¯n , l=1
Oν
˜ν |n|>K
τ d2 Kν −1 |hk, ω i | ≤ , 0 < |k| ≤ Kν ν−1 γ τ d2 Kν −1 ˜ν , |k| ≤ Kν , l ≤ Lν ≤ K k(hk, ων iI + A˜ν−1 ) k ≤ l γ 2 ξ ∈ Oν−1 : = τ d K k(hk, ων−1 iI + A˜lν−1 ⊗ I + I ⊗ A˜ν−1 )−1 k ≤ γν , |k| ≤ Kν , l, j ≤ Lν j 2 d |k| + |l − j| 6= 0, Kτ k(hk, ων−1 iI + A˜lν−1 ⊗ I − I ⊗ A˜ν−1 )−1 k ≤ γν , j |k| ≤ Kν , l, j ≤ Lν
where Lν is the smallest positive integer such that {Ω0n }|n|≤K˜ ν lie in Λ1 , · · · , ΛLν , and, ν−1 Al 0 ν−1 ˜ ν−1 < |n| ≤ K ˜ν. ˜ , Ω0n ∈ Λl , K Al = 0 Ω0n 4.1. Iteration Lemma. The preceding analysis may be summarized as follows. Lemma 4.1. If ε is sufficiently small, then the following holds for all ν = 0, 1, · · · . a) Hν is real analytic on Dν × Oν , Lν+1
Nν
= eν + hων (ξ), Ii +
X l=1
Pν
= P˘ν + P´0 + P`0 ,
hA˜νl zlν+1 , z¯lν+1 i +
X ˜ ν+1 |n|>K
Ω0n wn w ¯n ,
,
20
JIANSHENG GENG AND YINGFEI YI
and moreover, |ων+1 − ων |Oν ≤ c0 εν , kA˜ν+1 − A˜νl kOν ≤ c0 εν , l kXP ν kDν ,Oν ≤ εν , P˘ν = P˘ ν (θ, I, z ν , z¯ν , ξ) +
X
P˘nν (θ, I, z ν , z¯ν , ξ)wnαn w ¯nβn ,
˜ ν ,αn +βn ≥1 |n|>K
X
P´0 =
0 αm βm P´nm (ξ)wnαn w ¯nβn wm w ¯m ,
n6=m αn +βn ,αm +βm ≥1 αn +βn +αm +βm ≥3
P`0 =
X
O(|wn |3 ),
n
with ˜
kP˘nν (θ, I, z ν , z¯ν , ξ)kDν ,Oν ≤ e−(|n|−Kν ) , 0 kP´nm (ξ)kD ,O ≤ e−|n−m| . ν
ν
b) There is a symplectic transformation ˜ ν × Oν+1 → Dν Φν : D such that Hν+1 = Hν ◦ Φν . Proof. It is sufficient to verify the conditions C1)–C6) for all ν = 0, 1, · · · , which are easily seen to follow from the following conditions D1) rν −r2 ν+1 ln 15 ≤ Kν+1 ≤ 11 , 4
εν4
εν4τ d
1 4
4
D2) c0 γ −d (rν − rν+1 )−N εν ≤ 12 for all ν = 0, 1, · · · . We first let ε (hence ε0 ) be sufficiently small such that 4
ε0 < min{ where Ψ(r0 ) =
δ γ 5d r05N 1 , }, 5 9N +6 2 c0 Ψ(r0 ) 2
∞ Y
4 i
[(ri−1 − ri )−5N ]( 5 )
i=1
which is easily seen to be well-defined. Then 1
4
c0 γ −d (r0 − r1 )−N ε04 ≤
1 , 2
i.e., D2) holds for ν = 0. Recall that 1 8 1 ln ≤ K0 ≤ . 1 r0 ε 54 ε04τ d4 0 We see that D1) also holds for ν = 0. Now, for any ν ≥ 1, we have by induction that 1
4
4
5
4
1
4 c0 γ −d (rν − rν+1 )−N εν4 = c0 γ −d (rν − rν+1 )−N (4c0 γ −d (rν−1 − rν )−N εν−1 )4 5
4
1
4
1
5 ν
4 ≤ (24N +2 c50 γ −5d (rν−1 − rν )−5N εν−1 ) 4 ≤ (24N +2 c50 γ −5d Ψ(r0 )ε0 ) 4 ( 4 )
≤ (
r05N 25N +4
1
5 ν
)4(4) ≤
1 , 2
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
21
and 2 2ν+3 1 1 3ν 1 ln 5 ≤ ln ( 5 )ν ≤ ln ≤ 4ν K 0 rν − rν+1 εν4 r0 r ε 4 0 0 ε0 1 1 , = Kν+1 ≤ ( 6 )ν 1 ≤ 1 ε0 5 4τ d4 εν4τ d4 i.e., D1) and D2) hold true.
ν
= Φ0 ◦ Φ1 ◦ · · · ◦ Φν−1 , ν = 1, 2, · · · . Inductively, we have that
4.2. Convergence. Let Ψ ˜ ν × Oν+1 → D0 and Ψν : D
H0 ◦ Ψν = Hν = Nν + P ν for all ν = 1, 2, · · · . ˜ = ∩∞ Oν . Applying Lemma 4.1 and standard arguments (e.g. [22, 27]) we conclude that Let O ν=0 ˜ say to, H∞ , e∞ , N∞ , P ∞ , Ψ∞ , Hν , eν , Nν , P ν , Ψν , DΨν , ων converge uniformly on D( 21 r0 , 0) × O, ∞ DΨ , ω∞ respectively. It is clear that ∞ X ∞ ∞ N∞ = e∞ + hω∞ , Ii + hA∞ ¯l i. l zl , z l=1
Since
5
4
4
5 ν
4 ≤ (4c0 γ −d Ψ(r0 )ε0 )( 4 ) , εν = 4c0 γ −d (rν−1 − rν )−N εν−1 we have by Lemma 4.1 that XP ∞ |D( 1 r0 ,0)×O˜ ≡ 0. 2
Let ΦtH denote the flow of any Hamiltonian vector field XH . Since H0 ◦ Ψν = Hν , we have that ΦtH0 ◦ Ψν = Ψν ◦ ΦtHν .
(4.1)
The uniform convergence of Ψν , DΨν , XHν imply that one can pass the limit in the above to conclude that ΦtH0 ◦ Ψ∞ = Ψ∞ ◦ ΦtH∞ , ˜ It follows that on D( 1 r0 , 0) × O. 2
ΦtH0 (Ψ∞ (TN × {ξ})) = Ψ∞ ΦtN∞ (TN × {ξ}) = Ψ∞ (TN × {ξ}), ˜ Hence Ψ∞ (TN × {ξ}) is an embedded invariant torus of the original perturbed for all ξ ∈ O. ˜ The frequencies ω∞ (ξ) associated with Ψ∞ (TN × {ξ}) are slightly Hamiltonian system at ξ ∈ O. deformed from the unperturbed ones ω(ξ), and, the normal behaviors of the invariant tori Ψ∞ (TN × {ξ}) are governed by their respective normal frequency matrices A∞ l . 4.3. Measure estimates. For each ν = 1, 2, · · · , let Kντ d2 ν −1 ) , Rk (γ) = ξ ∈ Oν−1 : |hk, ων−1 i | > ( γ Kντ d2 −1 ) , ) k > ( Rνkl (γ) = ξ ∈ Oν−1 : k(hk, ων−1 iI + A˜ν−1 l γ Kτ 2 ν+ Rklj (γ) = {ξ ∈ Oν−1 : k(hk, ων−1 iI + A˜ν−1 ⊗ I + I ⊗ A˜ν−1 )−1 k > ( ν )d }, j l γ and, τ ˜ν−1 ⊗ I − I ⊗ A˜ν−1 )−1 k > ( Kν )d2 }, |k| + |l − j| 6= 0. Rν− j klj (γ) = {ξ ∈ Oν−1 : k(hk, ων−1 iI + Al γ Then [ [ [ [ [ Oν ⊆ Oν−1 \(( Rνk (γ)) ( Rνkl (γ)) ( Rν± klj (γ))), |k|≤Kν
˜ν |k|≤Kν ,l≤K
˜ν |k|≤Kν ,l,j≤K
22
JIANSHENG GENG AND YINGFEI YI
for all ν = 1, 2, · · · . Consider the resonant sets [ [ [ Rνk (γ)) ( Rν = (
Rνkl (γ))
[
˜ν |k|≤Kν ,l≤K
|k|≤Kν
[
(
Rν± klj (γ)).
˜ν |k|≤Kν ,l,j≤K
It is clear that ˜⊆ O\O
[
Rν .
ν≥1
Lemma 4.2. There is a constant C1 > 0 such that [ [ meas(Rνk (γ) Rνkl (γ) Rν± klj (γ)) ≤ C1
γ Kντ −1
˜ ν , and ν = 1, 2, · · · . for all |k| ≤ Kν , l, j ≤ K Proof. The proof follows from arguments in [31]. For simplicity, we only estimate the measures of ν+ ν ν Rν− klj (γ). Measure estimates for Rk , Rkl , and Rklj (γ) can be obtained similarly. We note that (4.2) hk, ων (ξ)iI + A˜ν−1 ⊗I −I ⊗ A˜ν−1 = hk, ων−1 (ξ)iI +diag(Ωn −Ωm , · · · , Ωn −Ωm )+W (ξ), j
l
1
1
r
r
where Ωn1 , Ωm1 , · · · , Ωnr , Ωmr are unperturbed frequencies independent of parameters, and dim(A˜ν−1 ⊗ I) = r ≤ d2 , and kW (ξ)kOν−1 ≤ ε0 . l In the case k = 0 and l 6= j, since by assumption (A2) |Ωnj − Ωmj | > γ, j = 1, · · · , r, we have 2 Kτ by the standard Neumann series expansion that kA˜ν−1 ⊗ I − I ⊗ A˜ν−1 )−1 k < 2 < ( ν+1 )d as j
l
γ
γ
ε 1, i.e., Rν− 0lj (γ) = ∅. In the case k 6= 0, we have by Lemma 5.7 that ˜ν−1 ⊗ I − I ⊗ A˜ν−1 )| < ( Rν2 klj ⊂ {ξ ∈ Oν−1 : | det(hk, ων−1 (ξ)iI + Al j
γ d2 τ −1 ) }. Kν
Denote g(ξ) = det(hk, ων−1 (ξ)iI + A˜ν−1 ⊗ I − I ⊗ A˜ν−1 ). j l Then it follows from (4.2) that r Y X g(ξ) = (hk, ων−1 (ξ)i + Ωni − Ωmi ) + aα (hk, ων−1 (ξ)i + Ωn − Ωm )α , α
i=1
where kaα kOν−1 ≤ ε0 , the multi-index α runs over the set α = (α1 , · · · , αr ), αi = 0 or 1, and Pr α i=1 i ≤ r − 1. Due to the choice of ε0 , we have that |∂ξr g(ξ)|
≥
r Y
|hk, ∂ξ ων (ξ)i| − ε0 |k|r−1 ≥ (δ − ε0 )|k|r − ε0 |k|r−1 ≥
i=1
Hence by Lemma 5.8, meas(Rν− klj ) ≤ C1 [(
δ r |k| . 2
γ d2 1 γ ) ] r ≤ C1 τ −1 . Kντ −1 Kν
Lemma 4.3. ˜ ≤ meas( meas(O \ O)
[
Rν ) = O(γ).
ν≥1
Proof. By Lemma 4.2, there exists a constant C2 > 0 such that [ X Rν± C1 meas( klj ) ≤ ˜ν |k|≤Kν ,l,j≤K
˜ν |k|≤Kν ,l,j≤K
≤ 4C1
X |k|≤Kν
γ Kντ −1
γ Kντ −3
≤ C2
γ Kντ −b−3
.
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
Similarly, there are constants C3 > 0, C4 > 0 such that [ meas( Rνk (γ) ≤ C3 |k|≤Kν
γ
[
meas(
Rνkl (γ)) ≤ C4
˜ν |k|≤Kν ,l≤K
,
Kντ −b−3
and,
23
γ . Kντ −b−3
Let τ ≥ b + 4. We have that ˜ ≤ meas( meas(O \ O)
[
Rν )
ν≥1
=
meas[
[
((
ν≥1
= O(
ν≥0
Rνk )
[
Kν+1
[
(
Rνkl )
[
˜ν |k|≤Kν ,l≤K
|k|≤Kν
γ
X
[
(
[
Rν± klj ))]
˜ν |k|≤Kν ,l,j≤K
) = O(γ).
This completes the measure estimate. 5. Appendix Lemma 5.1. kF GkD(r,s),O ≤ kF kD(r,s),O kGkD(r,s),O . Proof. Since (F G)klαβ = kF GkD(r,s),O
=
P
sup kwk<s kwk<s ¯
≤
sup kwk<s kwk<s ¯
≤
Fk−k0 ,l−l0 ,α−α0 ,β−β 0 Gk0 l0 α0 β 0 , we have X |(F G)klαβ |s2l |wα ||w ¯ β |e|k|r
k0 ,l0 ,α0 ,β 0
k,l,α,β
X
X
k,l,α,β
k0 ,l0 ,α0 ,β 0
|Fk−k0 ,l−l0 ,α−α0 ,β−β 0 Gk0 l0 α0 β 0 |s2l |wα ||w ¯ β |e|k|r
kF kD(r,s),O kGkD(r,s),O .
Lemma 5.2. (Generalized Cauchy inequalities) 1 kF kD(r,s) , σ 4 kFI kD(r, 21 s) ≤ 2 kF kD(r,s) , s 2 kFw kD(r, 12 s) ≤ kF kD(r,s) , s 2 kFw¯ kD(r, 12 s) ≤ kF kD(r,s) . s
kFθ kD(r−σ,s) ≤
Proof. See [27].
Let {·, ·} denote the Poisson bracket of smooth functions: X ∂F ∂G ∂F ∂G ∂F ∂G ∂F ∂G {F, G} = h , i−h , i+i ( − ). ∂I ∂θ ∂θ ∂I ∂wn ∂ w ¯n ∂w ¯n ∂wn n Lemma 5.3. There exists a constant c > 0 such that if kFn kD(r,s) < e−|n| , kGkD(r,s) < ε, then k{Fn , G}kD(r−σ, 12 s) < cσ −1 s−2 kFn kD(r,s) kGkD(r,s) ≤ cσ −1 s−2 εe−|n| .
24
JIANSHENG GENG AND YINGFEI YI
Proof. By Lemmas 5.1, 5.2, khFnI , Gθ ikD(r−σ, 21 s) khFnθ , GI ikD(r−σ, 12 s) X k Fnwm Gw¯m kD(r, 21 s)
< 4σ −1 s−2 kFn k · kGk, cσ −1 s−2 kFn k · kGk, X ≤ kFnwm kD(r, 21 s) kGw¯m kD(r, 12 s)
0 such that if kXF kD(r,s) < ε0 , kXG kD(r,s) < ε00 for some ε0 , ε00 > 0, then kX{F,G} kD(r−σ,ηs) < cσ −1 η −2 ε0 ε00 , 3
1
for any 0 < σ < r and 0 < η 1. In particular, if η ∼ ε 4 , ε0 ∼ ε, ε00 ∼ ε 4 , then 5
kX{F,G} kD(r−σ,ηs) ∼ ε 4 . Proof. See [16].
The following lemmas can be found in the Appendix of [12, 32].
Lemma 5.5. Let O be a compact set in RN for which small divisor conditions hold. Suppose that 2 2 ω(ξ) are C d Whitney-smooth functions in ξ ∈ O with derivative bounded by L and f (ξ) are C d d2 norm bounded by L. Then Whitney-smooth functions in ξ ∈ O with CW g(ξ) ≡
f (ξ) hk, ω(ξ)i
2
is C d Whitney-smooth in O with 4
4
τd kgkO < cγ −d K+ L.
Lemma 5.6. Let O be a compact set in RN for which small divisor conditions hold. Suppose 2 2 that Al (ξ), Rl (ξ) are respectively C d Whitney-smooth matrices and vectors, and ω(ξ) is a C d Whitney-smooth function with derivatives bounded by L. Then Fl (ξ) = M −1 Rl (ξ) 2
is C d Whitney-smooth with 4
4
τd kFl kO ≤ cγ −d K+ L, where M stands for either hk, ωiI + A˜l or hk, ωiI ± A˜l ⊗ I ± I ⊗ A˜j .
A KAM THEOREM FOR HAMILTONIAN NETWORKS WITH LONG RANGED COUPLINGS
25
Lemma 5.7. Let M be a r × r non-singular matrix with kM k < |k|. Then {ξ : kM −1 k > h} ⊆ {ξ : | det M |
0. If for some constant d > 0, |g (m) (u)| ≥ d for all u ∈ I, then 1 meas(Ih ) ≤ ch m , where c = 2(2 + 3 + · · · + m + d−1 ). References [1] S. Aubry, Anti–integrability in dynamical and variational problems, Physica D 86 (1995), 284–296 [2] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D 103 (1997), 201–250 [3] D. Bambusi, Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators, Nonlinearity 9 (1996), 433–457 [4] D. Bambusi and D. Vella, Quasi periodic breathers in Hamiltonian lattices with symmetries, Discrete Cont. Dyn. Sys. Ser. B 2 (2002), 389–399 [5] J. Bourgain, Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schr¨ odinger equations, Annals of Math. 148 (1998), 363–439 [6] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (1995), 629–639 [7] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE., Int. Math. Res. Notices (1994), 475–497 [8] J. Bourgain, Green’s function estimates for lattice Schr¨ odinger operators and applications. Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005 [9] J. Bourgain, Nonlinear Schr¨ odinger equations, Park City Series 5, American Mathematical Society, Providence, Rhode Island, 1999 [10] O. M. Braun and Y. S. Kivshar, Nonlinear dynamics of the Frenkel-Kontorova model, Phys. Reports 306 (1998), 1-108 [11] A. Carpio, L. L. Bonilla, Oscillatory wave fronts in chains of coupled nonlinear oscillators, Phys. Rev. E 67 (2003), 656-621 [12] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys. 211 (2000), 498-525 [13] W. Craig and C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure. Appl. Math. 46 (1993), 1409-1498 [14] T. Dauxois, M. Peyrard and C. R. Willis, Localised breather-like solution in a discrete Klein-Gordon model and application to DNA, Physica D 57 (1992), 267-282 [15] J. Fr¨ ohlich, T. Spencer and C. E. Wayne, Localization in disordered nonlinear dynamical systems, J. Stat. Phys. 42 (1986), 247–274 [16] J. Geng and J. You, A KAM theorem for one dimensional Schr¨ odinger equation with periodic boundary conditions, J. Differentail Equations 209(2005), 1-56 [17] J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Commun. Math. Phys. 262(2006), 343-372 [18] J. Geng and Y. Yi, Quasi-periodic solutions in a nonlinear Schr¨ odinger equation, submitted [19] M. Johansson and S. Aubry, Existence and stability of quasiperiodic breathers in the discrete nonlinear Schr¨ odinger equation, Nonlinearity 10 (1997), 1151–1178 [20] O. Kresse and L. Truskinovsky, Mobility of lattice defects: discrete and continuum approaches, J. Mech. Phys. Solids, 51 (2003), 1305-1332 [21] S. B. Kuksin, Hamiltonian perturbations of infinite–dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl. 21 (1987), 192-205 [22] S. B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, 1556, Springer, Berlin, 1993 [23] S. B. Kuksin and J. P¨ oschel, Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schr¨ odinger equation, Annals of Math. 143 (1996), 149-179 [24] Y. Li and Y. Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, Trans. Amer. Math. Soc. 357 No. 4 (2004), 1565-1600 [25] R. S. Mackay and S. Aubry, Proof of existence of breathers of time–reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7 (1994), 1623–1643 [26] J. P¨ oschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici 71 (1996), 269296
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[27] J. P¨ oschel, A KAM Theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 119-148 [28] J. P¨ oschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Comm. Math. Phys. 127 (1990), 351–393 [29] E. W. Prohofsky, K. C. Lu, L. L. Van Zandt, and B. F. Putnam, Breathing models and induced resonant melting of the double helix, Phys. Lett. A 70 (1979), 492-494 [30] C. E. Wayne, Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory, Commun. Math. Phys. 127 (1990), 479-528 [31] J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov’s non-resonance condition, J. Math. Pures Appl. 80 (2001), 1045-1067 [32] J. You, Perturbation of lower dimensional tori for Hamiltonian systems, J. Differential Equations 152 (1999), 1-29 [33] X. Yuan, Construction of quasi–periodic breathers via KAM technique, Commun. Math. Phys. 226 (2002), 61–100 [34] C. T. Zhang, Soliton excitations in deoxyribonucleic acid (DNA) double holices, Phys. Rev. A 35 (1987), 886-891 J. Geng: Department of Mathematics, Nanjing University, Nanjing 210093, P.R.China E-mail address: [email protected] (J. Geng) Y. Yi: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 E-mail address: [email protected] (Y. Yi)
