Divergent series resummations: examples in ODE s & PDE s Survey of works by G. Gentile (Roma3), V. Matropietro (Roma2), G.G. (Roma1) http://ipparco.roma1.infn.it 1
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
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
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
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
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 2 1. 5
Problem: construct a series representation of the solutions 6
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
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
(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
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
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
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
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
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
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
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
Simple self-energy graph (S.E., mass) R ν out ν in ν v ν v w R ν w = 0 17
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
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
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
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
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
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
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
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
Short Bibliography Low dimensional tori S.M. Graff: On the conservation for hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations 15 (1974), 1 69. 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), 1592 1596; A family of conditionally periodic solutions of a Hamiltonian systems, Soviet Math. Dokl. 9 (1968), 882 886. J. Moser: Convergent series expansions for almost periodic motions, Math. Ann. 169 (1967), 136 176. Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions, Math. Phys. Electronic J., <http:// 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), 151 167, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 533, Ed. C. Simy, Kluwer Academic Publishers, 1999. The bound of Siegel-Bryuno-Pöschel A.D. Bryuno: Analytic form of differential equa- 26
tions. The, II. (Russian), Trudy Moskov. Mat. Obšč. 25 (1971), 119 262; ibid. 26 (1972), 199 239. 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, 949 964, Ed. K. Osterwalder & R. Stora, North Holland, Amsterdam, 1986. Graph methods L. Chierchia, C. Falcolini: Compensations in small divisor problems, Comm. Math. Phys. 175(1996), 135 160. G. Gallavotti: Twistless KAM tori, Comm. Math. Phys. 164 nd (1994), no. 1, 145 156. 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, 343 411. 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, 101 140. J. Xu, J. You: Persistence of Lower Dimensional 27
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, 421 460. G. Gentile, Quasi-periodic solutions for two-level systems, Comm. Math. Phys. 242 (2003), no. 1 2, 221 250. G. Gentile, V. Mastropietro, Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method, in press. 28