Cantor families of periodic solutions for completely resonant nonlinear wave equations

By Massimiliano Berti, P. Bolle. 2004

82 citations recorded by the source corpus.

We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequencyω belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finitedimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity.

Subfields: Computer Networks and Communications, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/31d637fc6b03f4ed201b22ae04bc15c207760eb1

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In Computer Networks and Communications

1. Massimiliano Berti
2. Michela Procesi
3. P. Bolle
4. Berti M
5. M. E.
6. Bambusi D
7. Luca Biasco
8. Livia Corsi

In Statistical and Nonlinear Physics

1. Massimiliano Berti
2. Michela Procesi
3. P. Bolle
4. Luca Biasco
5. Livia Corsi
6. Berti M
7. M. E.
8. Bambusi D

Citations in the corpus

• Degenerate KAM theory for partial differential equations (2011, methodology): “We refer to [3] for a similar construction in the context of the Lyapunov–Schmidt approach for periodic solutions of PDEs.”
• An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds (2013, background): “The possibility to impose the non-resonance conditions through the “final solution” was yet observed in [3] (in a Lyapunov-Schmidt context) and in [2] for a KAM theorem.”
• Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions (2010, methodology): “Since such a f is analytic with respect to u, one could actually prove the existence of periodic solutions analytic in time following the techniques in [3].”
• KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation (2014, unknown): “This characterization of the set of parameters which fulfill all the required Melnikov non-resonance conditions (at any step of the iteration) was first observed in [6], [5] in an analytic setting.”
• Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs (2011, methodology): “We follow the cut-off procedure of [5].”