Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions
By G. Gentile, V. Mastropietro, Michela Procesi. 2004
63 citations recorded by the source corpus.
We consider the nonlinear string equation with Dirichlet boundary conditions utt−uxx=ϕ(u), with ϕ(u)=Φu3+O(u5) odd and analytic, Φ≠0, and we construct small amplitude periodic solutions with frequency ω for a large Lebesgue measure set of ω close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and non-resonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect to the nonlinear wave equations utt−uxx+Mu=ϕ(u), M≠0, is that not only the P equation but also the Q equation is infinite-dimensional.
Subfields: Numerical Analysis, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/8674c5eb70768dee4111ce416ad53526fa81e72a
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In Numerical Analysis
In Statistical and Nonlinear Physics
- Massimiliano Berti
- P. Bolle
- Michela Procesi
- Riccardo Montalto
- Ivar Ekeland
- Pierre Baldi
- Livia Corsi
- Luca Biasco
Citations in the corpus
- Cantor families of periodic solutions for completely resonant nonlinear wave equations (2004, methodology): “The article [15] applies to odd analytic nonlinearities like f (u) = au3 + O(u5) with a = 0 (the term u3 guarantees a nondegeneracy ∗Actually, [20] deals with the case of periodic boundary conditions in x (i.”