Skip to main content
SpringerLink
Log in

Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the nonlinear string equation with Dirichlet boundary conditions u tt −u xx =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 u tt −u xx +Mu=ϕ(u), M≠0, is that not only the P equation but also the Q equation is infinite-dimensional.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. New York: Dover, 1965

  2. Bambusi, D.: Lyapunov center theorem for some nonlinear PDE’s: a simple proof. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29(4), 823–837 (2000)

    Google Scholar 

  3. Bambusi, D., Paleari, S.: Families of periodic solutions of resonant PDEs. J. Nonlinear Sci. 11(1), 69–87 (2001)

    Google Scholar 

  4. Berti, M., Bolle, Ph.: Periodic solutions of nonlinear wave equations with general nonlinearities. Commun. Math. Phys. 243(2), 315–328 (2003)

    Google Scholar 

  5. Berti, M., Bolle, Ph.: Multiplicity of periodic solutions of nonlinear wave equations. Nonlinear Anal. 56, 1011–1046 (2004)

    Google Scholar 

  6. Birkhoff, G.D., Lewis, D.C.: On the periodic motions near a given periodic motion of a dynamical system. Ann. Math. 12, 117–133 (1933)

    Google Scholar 

  7. Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Internat. Math. Res. Notices 1994(11), 475–497 (1994)

    Google Scholar 

  8. Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995)

    Google Scholar 

  9. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. of Math. (2) 148(2), 363–439 (1998)

    Google Scholar 

  10. Bourgain, J.: Periodic solutions of nonlinear wave equations. In: Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Math. Chicago, IL: Univ. Chicago Press, 1999, pp. 69–97

  11. Brézis, H., Coron, J.-M., Nirenberg, L.: Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz. Commun. Pure Appl. Math. 33(5), 667–684 (1980)

    Google Scholar 

  12. Brézis, H., Nirenberg, L.: Forced vibrations for a nonlinear wave equation. Commun. Pure Appl. Math. 31(1), 1–30 (1978)

    Google Scholar 

  13. Craig, W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles. Panoramas and Syntheses 9. Paris: Société Mathématique de France, 2000

  14. Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math. 46, 1409–1498 (1993)

    Google Scholar 

  15. Craig, W., Wayne, C.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. 297–313

  16. Eliasson, L.H.: Absolutely convergent series expansions for quasi periodic motions. Math. Phys. Electron. J. 2, Paper 4, 33 pp. (electronic) (1996)

  17. Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45(2), 139–174 (1978)

    Google Scholar 

  18. Fröhlich, J., Spencer, T.: A rigorous approach to Anderson localization. In: Common trends in particle and condensed matter physics (Les Houches, 1983). Phys. Rep. 103(1-4), 9–25 (1984)

  19. Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164(1), 145–156 (1994)

    Google Scholar 

  20. Gallavotti, G., Gentile, G.: Hyperbolic low-dimensional invariant tori and summations of divergent series. Commun. Math. Phys. 227(3), 421–460 (2002)

    Google Scholar 

  21. Gallavotti, G., Gentile, G., Mastropietro, V.: A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems. J. Math. Phys. 40(12), 6430–6472 (1999)

    Google Scholar 

  22. Gentile, G.: Whiskered tori with prefixed frequencies and Lyapunov spectrum. Dynam. Stability Systems 10(3), 269–308 (1995)

    Google Scholar 

  23. Gentile, G., Mastropietro, V.: Construction of periodic solutions of the nonlinear wave equation with Dirichlet boundary conditions by the Lindstedt series method. J. Math. Pures Appl. 83(8), 1019–1065 (2004)

    Google Scholar 

  24. Godsil, C., Royle, G.: Algebraic graph theory. Graduate Texts in Mathematics 207. New York: Springer, 2001

  25. Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Sixth edition, San Diego: Academic Press, Inc., 2000

  26. Harary, F., Palmer, E.M.: Graphical enumeration. New York-London: Academic Press, 1973

  27. Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Mathematics 1556, Berlin: Springer, 1994

  28. Kuksin, S.B.: Fifteen years of KAM for PDE. In: Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 212, pp. 237–258, Amer. Math. Soc. Providence, RI, 2004

  29. Kuksin, S.B., Pöschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. of Math. (2) 143(1), 149–179 (1996)

    Google Scholar 

  30. Lidskii, B.V., Shul′man, E.I.: Periodic solutions of the equation u tt –u xx +u3=0. Funct. Anal. Appl. 22(4), 332–333 (1988)

    Google Scholar 

  31. Lyapunov, A.M.: Problème général de la stabilité du mouvement. Ann. Sc. Fac. Toulouse 2, 203–474 (1907)

    Google Scholar 

  32. Paleari, S., Bambusi, D., Cacciatori, S.: Normal form and exponential stability for some nonlinear string equations. Z. Angew. Math. Phys. 52(6), 1033–1052 (2001)

    Google Scholar 

  33. Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71(2), 269–296 (1996)

    Google Scholar 

  34. Rabinowitz, H.P.: Periodic solutions of nonlinear hyperbolic partial differential equations. Commun. Pure Appl. Math. 20, 145–205 (1967)

    Google Scholar 

  35. Rabinowitz, P.H.: Free vibrations for a semilinear wave equation. Commun. Pure Appl. Math. 31(1), 31–68 (1978)

    Google Scholar 

  36. Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990)

    Google Scholar 

  37. Weinstein, A.: Normal modes for nonlinear Hamiltonian systems. Invent. Math. 20, 47–57 (1973)

    Google Scholar 

  38. Whitney, H.: Analytic extensions of differential functions defined in closed sets. Trans. Amer. Math. Soc. 36(1), 63–89 (1934)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tre, 00146, Roma, Italy

    Guido Gentile

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133, Roma, Italy

    Vieri Mastropietro

  3. SISSA, 34014, Trieste, Italy

    Michela Procesi

Authors
  1. Guido Gentile
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vieri Mastropietro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michela Procesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by G. Gallavotti

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gentile, G., Mastropietro, V. & Procesi, M. Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions. Commun. Math. Phys. 256, 437–490 (2005). https://doi.org/10.1007/s00220-004-1255-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1255-8

Keywords

Search

Navigation