Quasi-periodic oscillations for wave equations under periodic forcing

Massimiliano Berti; Michela Procesi

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 2, page 109-116
  • ISSN: 1120-6330

Abstract

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Existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number.

How to cite

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Berti, Massimiliano, and Procesi, Michela. "Quasi-periodic oscillations for wave equations under periodic forcing." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.2 (2005): 109-116. <http://eudml.org/doc/252434>.

@article{Berti2005,
abstract = {Existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number.},
author = {Berti, Massimiliano, Procesi, Michela},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Nonlinear wave equation; Quasi-periodic solutions; Variational methods; Lyapunov-Schmidt reduction; Infinite dimensional Hamiltonian systems; nonlinear wave equation; quasi-periodic solutions; variational methods; infinite dimensional Hamiltonian systems},
language = {eng},
month = {6},
number = {2},
pages = {109-116},
publisher = {Accademia Nazionale dei Lincei},
title = {Quasi-periodic oscillations for wave equations under periodic forcing},
url = {http://eudml.org/doc/252434},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Berti, Massimiliano
AU - Procesi, Michela
TI - Quasi-periodic oscillations for wave equations under periodic forcing
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/6//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 2
SP - 109
EP - 116
AB - Existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number.
LA - eng
KW - Nonlinear wave equation; Quasi-periodic solutions; Variational methods; Lyapunov-Schmidt reduction; Infinite dimensional Hamiltonian systems; nonlinear wave equation; quasi-periodic solutions; variational methods; infinite dimensional Hamiltonian systems
UR - http://eudml.org/doc/252434
ER -

References

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  1. AMBROSETTI, A. - BADIALE, M., Homoclinics: Poincaré-Melnikov type results via a variational method. Annales I.H.P. - Analyse nonlin., v. 15, n. 2, 1998, 233-252. Zbl1004.37043MR1614571DOI10.1016/S0294-1449(97)89300-6
  2. BAMBUSI, D. - BERTI, M., A Birkhoof-Lewis type theorem for some Hamiltonian PDE's. SIAM Journal on Mathematical Analysis, to appear. Zbl1105.37045MR2176924DOI10.1137/S0036141003436107
  3. BENCI, V. - RABINOWITZ, P., Critical point theorems for indefinite functionals. Invent. Math., 52, n. 3, 1979, 241-273. Zbl0465.49006MR537061DOI10.1007/BF01389883
  4. BERTI, M. - BIASCO, L., Periodic solutions of nonlinear wave equations with non-monotone forcing terms. Rend. Mat. Acc. Lincei, s. 9, v. 16, 2005, 117-124. Zbl1225.35147MR2225505
  5. BERTI, M. - BOLLE, P., Periodic solutions of nonlinear wave equations with general nonlinearities. Comm. Math. Phys., 243, n. 2, 2003, 315-328. Zbl1072.35015MR2021909DOI10.1007/s00220-003-0972-8
  6. BERTI, M. - BOLLE, P., Cantor families of periodic solutions for completely resonant non linear wave equations. Preprint SISSA 2004. Zbl1103.35077
  7. BERTI, M. - PROCESI, M., Quasi-periodic solutions of completely resonant forced wave equations. Preprint SISSA. Zbl1100.35011MR2233048DOI10.1080/03605300500358129
  8. PLOTINIKOV, P.I. - YUNGERMANN, L.N., Periodic solutions of a weakly nonlinear wave equation with an irrational relation of period to interval length. Transl. in Diff. Eq., 24, n. 9, 1988, 1059-1065. MR965608
  9. PROCESI, M., Quasi-periodic solutions for completely resonant wave equations in 1D and 2D. Discr. Cont. Dyn. Syst., 13(3), August 2005, 541-552. Zbl1086.35007MR2152330DOI10.3934/dcds.2005.13.541
  10. RABINOWITZ, P., Periodic solutions of nonlinear hyperbolic partial differential equations. Comm. Pure Appl. Math., 20, 1967, 145-205. Zbl0152.10003MR206507

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