Divergent series resummations: examples in ODE s & PDE s

Transcription

1 Divergent series resummations: examples in ODE s & PDE s Survey of works by G. Gentile (Roma3), V. Matropietro (Roma2), G.G. (Roma1) 1

2 Hamiltonian rotators system α = ε α f(α, β), β = ε β f(α, β) α = (α 1,...,α r ), β = (β 1,...,β n r ), f: even trigonometric polynomial of deg. N, f = ν f ν(β)e iν α, ν Z r. Let f 0 (β) = 1 (2π) r f(α, β)dα 1

3 Special unperturbed motions ω = (ω 1,...,ω r ), ω ν > 1 C ν τ α = α 0 + ωt, β = β 0 r = n maximal tori or KAM tori. Are there motions of the same type in presence of interaction?? i.e. 2

4 Meaning of same type α = ψ + h(ψ), ψ ψ + ωt β = β 0 + k(ψ) α(t), β(t) is a solution? Equations (ω ψ ) 2 h(ψ )= ε α f(ψ + h(ψ ), β 0 + k (ψ )) (ω ψ ) 2 k (ψ )= ε β f(ψ +h(ψ ),β 0 +k(ψ )) h, k power series ε β f 0 (β 0 ) = 0 (ω ψ ) 2 k 1 (ψ) = β f(ψ, β 0 ) 0 = β f 0 (β 0 ) 3

5 Nonlinear wave equ. ( Klein Gordon ) u tt u xx +µu = f(u), f(u) = u 3 u(0,t) = u(π,t) = 0, µ > 0 if ω m = µ+m 2, do f = 0 periodic solutions like εcosω 1 t sinx, extend to f 0? u(x,t) = û n,m e inωt+imx, m,n Z with û n,m = û n, m = û n,m. 4

6 ûn,m[ Ω2 n 2 +ω 2 m] = ˆfn,m(u) ûn,m[ ω2 1 n2 + ω 2 m] = ν m(ε)ûn,m+ ˆfn,m(u), (1?) analytic (in ε) exponentially decaying (in m,n) solution given ω m verifying ω 1 n± ω m C 0 n τ, n 0;m,m 0,±1 ω 1 n±( ω m ± ω m ) C 0 n τ, (2?) for a solution of ω 2 m +ν m (ε) = ω 2 m so that Ω 2 = ω

7 Problem: construct a series representation of the solutions 6

8 Tori: r n. Lindstedt s series. Let ϑ be a tree with (1) p nodes v 0,...,v p 1, (2) one root r (3) possibly leaves with veinage. γ (4) γ v v Tree lines l = l v (v,v) joining nearest nodes(v,v)areorientedtowardstheroot. Total number of nodes k p 7

9 v γ η γ v η 3 ν 4 ν 2 ν 2 ν 1 η 0 ν 0 ν 1 r v 0 η 2 8

10 (a) Node v carries harmonic ν v Z r. (b) Line l = (v,v) carries current ν(l) def = w v ν w 0 (c) Current out of a leaf is 0. (d)linel = (v,v)carriespairη l = (γ v,γ v ), γ = 1,...,n (components). v γ γ v 9

11 Line l has a propagator matrix G l = δ γ,γ (ω ν(l)) 2 for leaf lines G l = δ γ,γ. γ γ 1 γ p Node v tensor F v = 1 p v! p v+1 γ v,γ v,1,...,γ v,p v f ν v (β 0 ) where γ = iν γ if γ r and β γ if γ > r. 10

12 Value, Reduced value, Leaf value Val(ϑ) γ = Val (ϑ) γ = v nodes(ϑ) v nodes(ϑ) F v v leaves(ϑ) F v v leaves(ϑ) L v l lines(ϑ) L v L v,γ = 2 βf 0 (β 0 ) 1 Val (ϑ L ), γ > r l l 0 G l Onlytherootlabelisnotcontracted: i.e. the value Val(ϑ) is a n vector G l, 11

13 Example: no leaves and r = n Val(ϑ) γ = l=(v v) ϑ ν v ν v (ω ν(l)) 2 v ϑ f ν v p v! The contractions of components labels generate the scalar products ν v ν v at each line and ν r has to be interpreted as the unit vector in the direction γ. 12

14 Lindstedt series The tori parametric equ. q (ψ )=(h(ψ ),k(ψ )), ψ T r q (k) ν,γ = ε k Val(ϑ) γ ϑ Θ k,ν,γ Θ k,ν,γ = (1) total number of nodes k (2) with root label γ (3) root line current ν (4) no line with 0 current(ext. Poincaré s th. on Lindstedt Newcomb series) 13

15 If r = n (maximal tori): no leaves; by KAM theorems series is convergent and indeed one can check that k th order BC k e κ ν 14

16 Siegel Bryuno Pöschel bound l v = (v v) has scale n = 0, 1, 2,... if 2 n 1 < ω ν(l v ) 2 n ν v ν v (ω ν(l v )) 2 4N 2 2 2n Val(ϑ) 1 p! (2N)2p F p 0 n= N n def = number of lines of scalen Recall 2 2nN n Val(ϑ) γ = l v =(v v) ϑ ν v ν v (ω ν(l v )) 2 v ϑ f ν v p v! 15

17 Val(ϑ) 1 p! (2N)2p F p 0 n= 2 2nN n If ν(l v ) ν(l w ) for all v > w N n an2 n/τ p 1 0 p! p!4p (2N) 2p F p ( 2 2naN2n/τ p ) = n= = 1 p! p!4p (2N) 2p F p (2 2aN 0 n2n/τ ) p = B p 16

18 Simple self-energy graph (S.E., mass) R ν out ν in ν v ν v w R ν w = 0 17

19 This is a self-energy subgraph if (1) the entering line and the exiting one have the same current ν, of scale n, and (2) all internal lines scale m > n+1 and their number is < a2 n/τ and (3) w R ν w = 0 Def: no S.E. sub-subgraph simple. Def: SE subgr. synonymous mass subgr. 18

20 Resummations of simple s.e. graphs Contribution to a tree value from a mass subgraph R inserted on line v v is ν v ν out (ω ν ) 2 ( w fν w pw! 1 (ω ν ) 2 ν MR (ν ) v (ω ν ) 2 ν v l ν w ν w (ω ν (l)) 2 ) ν in ν v (ω ν ) 2 w w R and l l R. 19

21 Let M 1 (ν) def = R ε R M R (ν), convergent sum because of the Siegel Bryuno Pöschel bound. Can insert m = 0,1,2,... mass subgr. on every line of a tree without SE m=0 =ν v 1 (ω ν) 2 ν v 1 (ω ν) 2 M 1 (ν) ν v ( M 1 (ν) (ω ν) 2 ) m ν v = 20

22 Cancellations BUT(ω ν) 2 M 1 (ν) = 0?? needm 1 (ν) = (ω ν) 2 m 1 ε(ν). Up to (1+O(ε 2 )) ( ) ω ν 2 M 1 = = ( (r r) ((n r) r) (r (n r)) ((n r) (n r)) (ω ν ) 2 i(ω ν )bε i( ω ν )bε = (ω ν ) 2 ε 2 β f 0 (β 0 ) ) 1 the r r KAM cancellations BUT (n r) (n r) elements can vanish on ornear theset of infinitelymany points ε for which (ω ν) 2 ε 2 β f 0(β 0 ) = 0. 21

23 eliminated simple SE subgraphs IF ε 2 β f 0(β 0 ) > 0. Elimination of overlapping graphs Define M 2 (ν) in the same way : considering trees with simple SE graphs at most and define value as in the preceding case making use, however, of the new propagators x 2 M(ν). Iterate indefinitely: G ε (k) (ν) ε (ν). k G( ) 22

24 Invariant torus equ.: just consider all graphs without SE and new propagator ((ω ν) 2 M 1 def (ν)) = G ( ) (ν) by Siegel Bryuno Pöschel no converg. prob.: hence algorithm for LNP series. 23

25 If ε > 0 and β 0 is a maximum the popagator matrix has no 0 eigenvalue. Hence no difference from maximal case. Convergence in complex ε domain D where (ω ν) 2 ε 2 β f 0(β 0 ) γ(ω ν) 2. complex ε plane O Fig.3: Lower dimensional tori analyt. domain D 0 tori. Cusp at the origin is of ond order. 24

26 complex ε plane Fig.4: Can D 0 in Fig.3 be extended? Perhaps be (near the origin) as in the picture? Real axis reached in cusps with apex at a set I; for ε I the parametric eq. correspond to elliptic tori which would be analytic continuations of the hyperbolic tori. 25

27 Short Bibliography Low dimensional tori S.M. Graff: On the conservation for hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations 15 (1974), V.K. Mel nikov: On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function, Soviet Math. Dokl. 6(1965), ; A family of conditionally periodic solutions of a Hamiltonian systems, Soviet Math. Dokl. 9 (1968), J. Moser: Convergent series expansions for almost periodic motions, Math. Ann. 169 (1967), Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions, Math. Phys. Electronic J., < mpej.unige.ch>, 2(1996), Paper 4 A. Jorba, R. Llave, M. Zou: Lindstedt series for lower dimensional tori, in Hamiltonian systems with more than two degrees of freedom (S Agary, 1995), , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 533, Ed. C. Simy, Kluwer Academic Publishers, The bound of Siegel-Bryuno-Pöschel A.D. Bryuno: Analytic form of differential equa- 26

28 tions. The, II. (Russian), Trudy Moskov. Mat. Obšč. 25 (1971), ; ibid. 26 (1972), J. Pöschel: Invariant manifolds of complex analytic mappings near fixed points, in Critical Phenomena, Random Systems, Gauge Theories, Les Houches, Session XLIII (1984), Vol. II, , Ed. K. Osterwalder & R. Stora, North Holland, Amsterdam, Graph methods L. Chierchia, C. Falcolini: Compensations in small divisor problems, Comm. Math. Phys. 175(1996), G. Gallavotti: Twistless KAM tori, Comm. Math. Phys. 164 nd (1994), no. 1, Twistless KAM tori, quasiflat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review, Rev. Math. Phys. 6 (1994), no. 3, More recent papers J. Bricmont, A. Kupiainen, A. Schenkel: Renormalization group for the Melnikov problem for PDE s, Comm. Math. Phys. 221 (2001), no. 1, J. Xu, J. You: Persistence of Lower Dimensional 27

29 Tori Under the First Melnikov s Non-resonance Condition, Nanjing University Preprint, 1 21, 2001, J. Math. Pures Appl., in press. Xiaoping Yuan: Construction of quasi-periodic breathers via KAM techniques Comm. Math. Phys., in press. G. Gallavotti, G. Gentile, Hyperbolic low-dimensional tori and summations of divergent series, Comm. Math. Phys. 227 (2002), no. 3, G. Gentile, Quasi-periodic solutions for two-level systems, Comm. Math. Phys. 242 (2003), no. 1 2, G. Gentile, V. Mastropietro, Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method, in press. 28