We study small amplitude solutions of the nonlinear wave equation utt - ucursive Greek chicursive Greek chi = ψ(u) u(0, t) = 0 = u(π, t) (0.1) with an analytic nonlinearity of the type ψ(u) = ±u2k-1 + script O sign(u2k) , k ≥ 2 . For this system we introduce a new method to compute the resonant normal form obtaining a simple formula for it. Then we specialize to the case k = 2 and find all periodic solutions of the averaged system and also of what we call the "simplified system", namely a Hamiltonian system that we will prove to approximate 0.1 up to an error exponentially small with the energy of the initial datum. Furthermore, for one of such orbits we prove a strong (Nekhoroshev type) stability property with respect to the complete dynamics.
Normal form and exponential stability for some nonlinear string equations / S. Paleari, D. Bambusi, S. Cacciatori. - In: ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK. - ISSN 0044-2275. - 52:6(2001 Nov), pp. 1033-1052. [10.1007/PL00001582]
Normal form and exponential stability for some nonlinear string equations
S. PaleariPrimo
;D. BambusiSecondo
;
2001
Abstract
We study small amplitude solutions of the nonlinear wave equation utt - ucursive Greek chicursive Greek chi = ψ(u) u(0, t) = 0 = u(π, t) (0.1) with an analytic nonlinearity of the type ψ(u) = ±u2k-1 + script O sign(u2k) , k ≥ 2 . For this system we introduce a new method to compute the resonant normal form obtaining a simple formula for it. Then we specialize to the case k = 2 and find all periodic solutions of the averaged system and also of what we call the "simplified system", namely a Hamiltonian system that we will prove to approximate 0.1 up to an error exponentially small with the energy of the initial datum. Furthermore, for one of such orbits we prove a strong (Nekhoroshev type) stability property with respect to the complete dynamics.Pubblicazioni consigliate
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