Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type
By Pierre Baldi. 2012
57 citations recorded by the source corpus.
We prove the existence of time-periodic, small amplitude solutions of autonomous quasilinear or fully nonlinear completely resonant pseudo-PDEs of Benjamin-Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero. At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions). The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation that contains derivatives of the highest order. The main difficulties of the problem are: an infinite-dimensional bifurcation equation, and small divisors in the linearized operator, where also the highest order derivatives have nonconstant coefficients. The main technical step of the proof is the reduction of the linearized operator to constant coefficients up to a regularizing rest, by means of changes of variables and conjugation with simple linear pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov and Toland for standing water waves (ARMA 2005). Other ingredients are a suitable Nash-Moser iteration in Sobolev spaces, and Lyapunov-Schmidt decomposition. (Version 2: small change in Section 2).
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/4fda694e6c2a96bc5f105d2a7b64ef9b6420b750
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Mathematical Physics
- Riccardo Montalto
- Michela Procesi
- Massimiliano Berti
- Pierre Baldi
- E. Haus
- R. Feola
- Filippo Giuliani
- P. Bolle
In Statistical and Nonlinear Physics
- Massimiliano Berti
- Riccardo Montalto
- Michela Procesi
- Pierre Baldi
- E. Haus
- R. Feola
- Filippo Giuliani
- P. Bolle
Citations in the corpus
- Reducibility of first order linear operators on tori via Moser's theorem (2018, methodology): “,G | {z } ×r2 , I,...,I | {z } ×k−r1−r2 ]. The L2 norms of the summands above are bounded by 2Ck,r1,r2kukk|G|h1,∞ ...|G|hr 1,∞|G| r2 ∞. Then one can follow exactly the same proof of Lemma B.4-(ii) in [3]. In order to prove (A.7) we use formula (6.15) of [5] which reads in terms of the Lipschitz semi”
- KAM for autonomous quasi-linear perturbations of mKdV (2014, result): “E-mail addresses: pietro.baldi@unina.it (P. Baldi), berti@sissa.it (M. Berti), riccardo.montalto@sissa.it, riccardo.montalto@math.uzh.ch (R. Montalto). http://dx.doi.org/10.1016/j.anihpc.2015.07.003 0294-1449/© 2015 Elsevier Masson SAS.”
- Time quasi-periodic gravity water waves in finite depth (2017, background): “5 in [6] (and Appendices H and I in [42] for similar statements).”