![Introduction Main results Ingredients of the proof
Topology: Sobolev space of sequences
For any s ≥ 0, σ ≥ 0 define
(X, Y )
2
s,σ :=
1
N
N−1
k=1
[k]
2s
N e2σ[k]N
ωk
|Xk |2
+ |Yk |2
2
,
where [k]N := min(|k|, |N − k|).
Let Bs,σ
(R) the ball or center 0 and radius R in this topology.
Remarks:
- (X, Y )
2
0,0 ∼
Energy
N
- σ > 0: exponentially localized states](https://image.slidesharecdn.com/presentazionetoda-160206123839/85/Birkhoff-coordinates-for-the-Toda-Lattice-in-the-limit-of-infinitely-many-particles-with-application-to-FPU-8-320.jpg)
![Introduction Main results Ingredients of the proof
Lax matrices
Flaschka coordinates: bj = −pj , aj = e
1
2 (qj −qj+1)
− 1, not canonical.
L(b, a) =
b0 1 + a0 0 . . . 1 + aN−1
a0 b1 1 + a1
...
.
.
.
0 1 + a1 b2 . . . 0
.
.
.
...
...
... 1 + aN−2
1 + aN−1 . . .
... 1 + aN−2 bN−1
,
The equations of motion are equivalent to: d
dt L = [B, L]
Eigenvalues of L are conserved quantities λj (b(t), a(t)) = λj (b0, a0)
Poisson commute {λj , λi } = 0](https://image.slidesharecdn.com/presentazionetoda-160206123839/85/Birkhoff-coordinates-for-the-Toda-Lattice-in-the-limit-of-infinitely-many-particles-with-application-to-FPU-13-320.jpg)
![Introduction Main results Ingredients of the proof
Spectral perturbation
(1) Spectral projectors on the eigenspaces of L(b, a)
Ej (b, a) subspace associated to λ2j−1(b, a), λ2j (b, a).
Pj (b, a) spectral projection on Ej (b, a):
Pj (b, a) = −
1
2πi Γj
(L(b, a) − λ)
−1
dλ,
(2) Transformation operator
Uj (b, a) = 1 − (Pj (b, a) − Pj0)
− 1
2 Pj (b, a)
maps isometrically the unperturbed eigenspace into the perturbed one
ImUj (b, a) = Ej (b, a) , Uj (b, a)f C2N = f C2N if f ∈ Ej0
(3) define
fj (b, a) = Uj (b, a)fj0
zj (b, a) = 2
N
ω j
N
−1/2
L(b, a) − λ
0
2j fj (b, a), fj (b, a)
(4) prove that zj j
’s (as functions of (X, Y )) satisfy Kuksin-Perelman assumptions. In particular for b, a real
one has
|zj (b, a)|
2
= 2
N
ω j
N
−1
γ
2
j (b, a)
Apply Kuksin-Perelman with weights w2
j = N4
[j]2s
N e2σ[j]N 2
N
ω j
N
.](https://image.slidesharecdn.com/presentazionetoda-160206123839/85/Birkhoff-coordinates-for-the-Toda-Lattice-in-the-limit-of-infinitely-many-particles-with-application-to-FPU-16-320.jpg)
Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with application to FPU
- 1. Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU D. Bambusi, A. Maspero September 4, 2014
- 2. Introduction Main results Ingredients of the proof Overview Toda’s chain with N particles: H(p, q) = 1 2 N−1 j=0 p2 j + N−1 j=0 eqj −qj+1 periodic boundary conditions: pj+N = pN , qj+N = qj phase space: p, q ∈ RN : N−1 j=0 pj = N−1 j=0 qj = 0 . symplectic form: j dpj ∧ dqj Definition: Let H(p, q) be a Hamiltonian. Birkhoff coordinates are canonical coordinates (x, y) such that the Hamiltonian in the new coordinates is a function only of the actions H(x, y) = H (Ij ) , Ij = x2 j + y2 j 2 Φ : (p, q) → (x, y) is called Birkhoff map Aim: study the domain of analyticity of BC when N → ∞
- 3. Introduction Main results Ingredients of the proof Overview Toda’s chain with N particles: H(p, q) = 1 2 N−1 j=0 p2 j + N−1 j=0 eqj −qj+1 periodic boundary conditions: pj+N = pN , qj+N = qj phase space: p, q ∈ RN : N−1 j=0 pj = N−1 j=0 qj = 0 . symplectic form: j dpj ∧ dqj Definition: Let H(p, q) be a Hamiltonian. Birkhoff coordinates are canonical coordinates (x, y) such that the Hamiltonian in the new coordinates is a function only of the actions H(x, y) = H (Ij ) , Ij = x2 j + y2 j 2 Φ : (p, q) → (x, y) is called Birkhoff map Aim: study the domain of analyticity of BC when N → ∞
- 4. Introduction Main results Ingredients of the proof Motivation Dynamic of chains with many particles: metastability in FPU HFPU (p, q) = N−1 j=0 p2 j 2 + (qj − qj+1)2 2 + (qj − qj+1)3 6 + β (qj − qj+1)4 24 Long time behaviour of (specific) energies of normal modes: Ek := 1 N |ˆpk |2 + ω k N 2 |ˆqk |2 2 , 1 ≤ k ≤ N − 1 , and of their time average Ek (t) := 1 t t 0 Ek (s)ds FPU: initial data with E1(0) = 0 and Ek (0) = 0 ∀k = 1, N − 1.
- 5. Introduction Main results Ingredients of the proof Ek (t) present a recurrent behaviour, Ek (t) quickly relax to a sequence ¯Ek exponentially decreasing with k. - Benettin, Ponno ’11: numerical analysis. Phenomenon persists in the thermodynamic limit if initial phases are randomly chosen. Question: Stability of metastable packet in Toda?
- 6. Introduction Main results Ingredients of the proof Related results In the regime Energy N ∼ 1 N4 - Bambusi, Ponno ’06: analytical analysis. Persistence of the metastable packet in FPU for long times using KdV approximation. - Bambusi, Kappeler, Paul ’13: spectral data Toda converges to spectral data of KdV
- 7. Introduction Main results Ingredients of the proof Birkhoff map for Toda’s chain Linear Birkhoff variables Xk , Yk in which the quadratic part of HToda is H0 = N−1 k=1 ωk X2 k + Y 2 k 2 , ωk := 2 sin kπ 2N (1) Theorem (Henrici, Kappeler ’08) For any integer N ≥ 2 there exists a global real analytic symplectic diffeomorphism ΦN : RN−1 × RN−1 → RN−1 × RN−1 , (X, Y ) = ΦN (x, y) with the following properties: (i) The Hamiltonian HToda ◦ ΦN is a function of the actions Ik := x2 k +y2 k 2 only, i.e. (xk , yk ) are Birkhoff variables for the Toda Lattice. (ii) The differential of ΦN at the origin is the identity: dΦN (0, 0) = 1. Main goal: Analyticity properties of the map ΦN as N → ∞. ΦN is globally analytic but it might (and actually does!) develop singularities as N → ∞.
- 8. Introduction Main results Ingredients of the proof Topology: Sobolev space of sequences For any s ≥ 0, σ ≥ 0 define (X, Y ) 2 s,σ := 1 N N−1 k=1 [k] 2s N e2σ[k]N ωk |Xk |2 + |Yk |2 2 , where [k]N := min(|k|, |N − k|). Let Bs,σ (R) the ball or center 0 and radius R in this topology. Remarks: - (X, Y ) 2 0,0 ∼ Energy N - σ > 0: exponentially localized states
- 9. Introduction Main results Ingredients of the proof Main theorem Theorem (Bambusi, M.) For any s ≥ 1, σ ≥ 0 there exist strictly positive constants Rs,σ,Rs,σ, such that for any N 1, the map ΦN is real analytic as a map from Bs,σ (Rs,σ/N2 ) to Bs,σ (Rs,σ/N2 ). The same holds for the inverse map Φ−1 N possibly with a different Rs,σ. Remark: - We work in the regime Energy N ∼ 1 N4 - Stability of metastable packett of Toda for all times
- 10. Introduction Main results Ingredients of the proof Application to FPU Theorem Consider the FPU system. Fix s ≥ 1 and σ ≥ 0; then there exist constants R0, C2, T, such that the following holds true. Consider a real initial datum such that E1(0) ≤ R N4 , R ≤ R0 Ek (0) = 0 , k = 1 . Then, along the corresponding solution, one has Ek (t) ≤ R N4 e−2σk , ∀ 1 ≤ k ≤ N/2 , for |t| ≤ TN4 R2 · 1 |β − 1| + C2R 1 N2 .
- 11. Introduction Main results Ingredients of the proof Keys ingredients in the proof Construction of ΦN : Vey type theorem for infinite dimensional systems (Kuksin, Perelman ’10); perturbative analysis of spectral theory of Lax operator of Toda;
- 12. Introduction Main results Ingredients of the proof Kuksin-Perelman theorem ’10 2 w := (ξj )j≥1, ξj ∈ C : j w2 j |ξj |2 < ∞ , as a real Hilbert space. Let ξ, η = Re j ξj ηj , ω0(ξ, η) = iξ, η . Theorem Assume that there exists a real analytic map Ψ : Bw (ρ) → 2 w , ρ > 0, with (i) dΨ(0) = 1 (ii) Ψ = (ψj )j≥1 fulfills: |ψj |2 , j ≥ 1, are in involution; (iii) analyticity of Cauchy majorants of Ψ and dΨ∗ Then there exists Φ = (φj )j≥1, Φ : Bw (ρ ) → 2 w , such that (i), (ii), (iii) hold and 1 the foliation in a neighborhood of the origin in 2 w defined by |ψj | 2 = cj , ∀j is the same foliation as |φj | 2 = cj , ∀j 2 Φ∗ ω0 = ω0; namely {φj }j are canonical coordinates.
- 13. Introduction Main results Ingredients of the proof Lax matrices Flaschka coordinates: bj = −pj , aj = e 1 2 (qj −qj+1) − 1, not canonical. L(b, a) = b0 1 + a0 0 . . . 1 + aN−1 a0 b1 1 + a1 ... . . . 0 1 + a1 b2 . . . 0 . . . ... ... ... 1 + aN−2 1 + aN−1 . . . ... 1 + aN−2 bN−1 , The equations of motion are equivalent to: d dt L = [B, L] Eigenvalues of L are conserved quantities λj (b(t), a(t)) = λj (b0, a0) Poisson commute {λj , λi } = 0
- 14. Introduction Main results Ingredients of the proof Jacobi matrix spectrum and spectral gaps Theorem For (b, a) ∈ RN × RN >−1, the spectrum of L± is a sequence of eigenvalues satisfying the relation λ0 < λ1 ≤ λ2 < λ3 ≤ λ4 < . . . < λ2N−3 ≤ λ2N−2 < λ2N−1. If N is even λ0, λ3, λ4, . . . , λ2N−1 are eigenvalues of L+ , while the others are eigenvalues of L− . Viceversa if N is odd. Definition The j-th spectral gap is γj := λ2j − λ2j−1 Fact: 1 Battig, Gr´ebert, Guillot, Kappeler ’93 the sequence of gaps determine all the spectrum; 2 {γ2 j (b, a)}j are set of complete integrals, real analytic and in involution
- 15. Introduction Main results Ingredients of the proof Perturbative analysis L(b, a) = 0 1 0 . . . 1 1 0 1 ... . . . 0 1 0 . . . 0 . . . ... ... ... 1 1 . . . . . . 1 0 L0 + b0 a0 0 . . . a2N−1 a0 b1 a1 ... . . . 0 a1 b2 . . . 0 . . . ... ... ... a2N−2 a2N−1 . . . . . . a2N−2 b2N−1 Lp:perturbation Eigenvalues L0: λ0 0 = −2; λ0 2l−1 = λ0 2l = −2 cos lπ N ; λ0 2N−1 = 2 Distance of unperturbed eigenvalues: λ0 2l − λ0 2j ∼ l2 − j2 N2 Aim: reconstruct the spectral data of L from the spectral data of L0.
- 16. Introduction Main results Ingredients of the proof Spectral perturbation (1) Spectral projectors on the eigenspaces of L(b, a) Ej (b, a) subspace associated to λ2j−1(b, a), λ2j (b, a). Pj (b, a) spectral projection on Ej (b, a): Pj (b, a) = − 1 2πi Γj (L(b, a) − λ) −1 dλ, (2) Transformation operator Uj (b, a) = 1 − (Pj (b, a) − Pj0) − 1 2 Pj (b, a) maps isometrically the unperturbed eigenspace into the perturbed one ImUj (b, a) = Ej (b, a) , Uj (b, a)f C2N = f C2N if f ∈ Ej0 (3) define fj (b, a) = Uj (b, a)fj0 zj (b, a) = 2 N ω j N −1/2 L(b, a) − λ 0 2j fj (b, a), fj (b, a) (4) prove that zj j ’s (as functions of (X, Y )) satisfy Kuksin-Perelman assumptions. In particular for b, a real one has |zj (b, a)| 2 = 2 N ω j N −1 γ 2 j (b, a) Apply Kuksin-Perelman with weights w2 j = N4 [j]2s N e2σ[j]N 2 N ω j N .
- 17. Introduction Main results Ingredients of the proof Theorem (Bambusi, M.) Assume that for some s ≥ 1, σ ≥ 0 there exist strictly positive R, R and α ≥ 0, Ns,σ ∈ N, s.t., for any N ≥ Ns,σ, the map ΦN is analytic in the complex ball Bs,σ (R/Nα ) and fulfills ΦN Bs,σ R Nα ⊂ Bs,σ R Nα , (2) then one has α ≥ 2. Theorem (Bambusi, M.) For any s ≥ 1, σ ≥ 0 there exist strictly positive R, C, Ns,σ ∈ N, such that, for any N ≥ Ns,σ, α ∈ R, the quadratic form d2 ΦN (0, 0) fulfills sup v∈Bs,σ R ( R Nα ) d2 ΦN (0, 0)(v, v) s,σ ≥ CR2 N2−2α . (3) As N → ∞, the real diffeomorphism ΦN develops a singularity at zero in the second derivative.
- 18. Introduction Main results Ingredients of the proof Proof of the inverse theorem (1) Taylor expansion of ΦN at the origin: ΦN = 1 + QΦN + O( ξ, η 3 s,σ) , QΦN is a quadratic polynomial and Hamiltonian vector field: QΦn = XχΦN χΦN cubic complex valued polynomial. (2) Taylor expansion of HToda ◦ ΦN at the origin HToda ◦ ΦN = H0 + H0, χΦN + H1 + h.o.t. ΦN Birkhoff map ⇒ HToda ◦ ΦN in BNF ⇒ only terms of even degree. H0, χΦN + H1 = 0 If χΦN = |K|+|L|=3 χK,L ξK ηL, then χK,L = HK,L −iω·(K−L) (3) Compute explicitly! For ¯v = ((ξ1, 0, 0, ..., 0), (¯ξ1, 0, 0, ..., 0)) 2ω1 − ω2 ∼ 1 N3 =⇒ QφN (¯v) s,σ ≥ N2 ¯v s,σ