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Communications in Mathematical Physics

, Volume 243, Issue 2, pp 315–328 | Cite as

Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities

  • Massimiliano BertiEmail author
  • Philippe Bolle
Article

Abstract

We present a variational principle for small amplitude periodic solutions, with fixed frequency, of a completely resonant nonlinear wave equation. Existence and multiplicity results follow by min-max variational arguments.

Keywords

Wave Equation Periodic Solution Variational Principle Small Amplitude Nonlinear Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ambrosetti, A., Badiale, M.: Homoclinics: Poincaré-Melnikov type results via a variational approach. Annales I. H. P. - Analyse nonlin. 15(2), 233–252 (1998)zbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Rabinowitz, P.: Dual Variational Methods in Critical Point Theory and Applications. J. Func. Anal. 14, 349–381 (1973)zbMATHGoogle Scholar
  3. 3.
    Bambusi, D.: Lyapunov Center Theorems for some nonlinear PDEs: A simple proof. Ann. Sc. Norm. Sup. di Pisa, Ser. IV XXIX, fasc. 4, (2000)Google Scholar
  4. 4.
    Bambusi,~D.: Families of periodic solutions of reversible PDEs. Preprint available at http://www.math.utexas.edu/mparcGoogle Scholar
  5. 5.
    Bambusi, D., Paleari, S.: Families of periodic solutions of resonant PDEs. J. Nonlinear Sci. 11, 69–87 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bambusi, D., Cacciatori, S., Paleari, S.: Normal form and exponential stability for some nonlinear string equations. Z. Angew. Math. Phys. 52(6), 1033–1052 (2001)zbMATHGoogle Scholar
  7. 7.
    Bambusi, D., Paleari, S.: Families of periodic orbits for some PDE’s in higher dimensions. Comm. Pure and Appl. Analysis 1(4) (2002)Google Scholar
  8. 8.
    Berti, M., Bolle, P.: Multiplicity of periodic solutions of nonlinear wave equations, to appear in Nonlinear Analysis: Theory, Methods, & ApplicationsGoogle Scholar
  9. 9.
    Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. 148, 363–439 (1998)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Craig, W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles. Panoramas et Synthèses, 9, Paris: Société Mathématique de France, 2000Google Scholar
  11. 11.
    Craig, W., Wayne, E.: Newton’s method and periodic solutions of nonlinear wave equation. Commun. Pure and Appl. Math. XLVI, 1409–1498 (1993)Google Scholar
  12. 12.
    Craig, W., Wayne, E.: Nonlinear waves and the 1:1:2 resonance. In: Singular limits of dispersive waves (Lyon, 1991), NATO Adv. Sci. Inst. Ser. B Phys., 320, New York: Plenum, 1994, pp. 297–313Google Scholar
  13. 13.
    Fadell, E.R., Rabinowitz, P.: Generalized cohomological index theories for the group actions with an application to bifurcations question for Hamiltonian systems. Inv. Math. 45, 139–174 (1978)zbMATHGoogle Scholar
  14. 14.
    Kuksin, S.B.: Perturbation of conditionally periodic solutions of infinite-dimensional Hamiltonian systems. Izv. Akad. Nauk SSSR, Ser. Mat. 52(1), 41–63 (1988)Google Scholar
  15. 15.
    Moser, J.: Periodic orbits near an Equilibrium and a Theorem by Alan Weinstein. Commun. Pure Appl. Math. XXIX (1976)Google Scholar
  16. 16.
    Weinstein, A.: Normal modes for Nonlinear Hamiltonian Systems. Inv. Math. 20, 47–57 (1973)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.S.I.S.S.A.TriesteItaly
  2. 2.Université d’AvignonAvignonFrance

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