Borel Summability and Lindstedt Series

Transseries and Écalle-Borel summability

ar X iv : m at h - ph / 0 60 10 32 v 1 1 5 Ja n 20 06 Borel summability and Lindstedt series

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining C functions of the perturbation strength: here we find sufficient conditions for the Borel summability of their sums in t...

Lindstedt Series and Kolmogorov Theorem

the KAM theorem from a combinatorial viewpoint. Lindstedt, Newcomb and Poincaré introduced a remarkable trigonometric series motivated by the analysis of the three body problem, [P]. In modern language, [G2], it is the generating function, that I call Lindstedt series here, ~h(~ ψ) = ∑∞ k=1 ε k~h(k)(~ ψ) of the sequence ~h(k)(~ ψ) of trigonometric polynomials associated with well known combinat...

Lindstedt Series, Ultraviolet Divergences and Moser's Theorem

Moser's invariant tori for a class of nonanalytic quasi integrable even hamiltonian systems are shown to be analytic in the perturbation parameter. We do so by exhibiting a summation rule for the divergent series (\Lindstedt series") that formally deene them. We nd additional cancellations taking place in the formal series, besides the ones already known and necessary in the analytic case (i.e....

: Existence and Uniqueness of Solutions , Asymptotic and Borel Summability Properties

We consider a system of n-th order nonlinear quasilinear partial differential equations of the form ut + P(∂ j x)u + g ( x, t, {∂ xu} ) = 0; u(x, 0) = uI(x) with u ∈ C, for t ∈ (0, T ) and large |x| in a poly-sector S in C (∂ x ≡ ∂ j1 x1∂ j2 x2 ...∂ jd xd and j1 + ... + jd ≤ n). The principal part of the constant coefficient n-th order differential operator P is subject to a cone condition. The...