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Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs

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Abstract

We consider infinite dimensional Hamiltonian systems. We prove the existence of “Cantor manifolds” of elliptic tori–of any finite higher dimension–accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are “branching” points of other Cantor manifolds of higher dimensional tori. We also answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a pre-assigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation.

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References

  1. Bambusi D.: Birkhoff normal form for some nonlinear PDEs. Commun. Math. Phys. 234(2), 253–285 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Bambusi D., Berti M.: A Birkhoff-Lewis type theorem for some Hamiltonian PDE’s. SIAM J. Math. Anal. 37(1), 83–102 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bambusi D., Berti M., Magistrelli E.: Degenerate KAM theory for partial differential equations. J. Diff. Eqs. 250(8), 3379–3397 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bambusi D., Grébert B.: Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J. 135(3), 507–567 (2005)

    Article  Google Scholar 

  5. Berti M., Bolle P.: Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math. J. 134(2), 359–419 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berti M., Bolle P., Procesi M.: An abstract Nash-Moser Theorem with parameters and applications to PDEs. Ann. Inst. H. Poincaré, Anal. Nonlin. 27, 377–399 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Biasco L., Di Gregorio L.: A Birkhoff-Lewis type Theorem for the nonlinear wave equation. Arch. Rat. Mech. 196(1), 303–362 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bobenko A., Kuksin S.: The nonlinear Klein-Gordon equation on an interval as a perturbed sine-Gordon equation. Comment. Math. Helv. 70(1), 63–112 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bourgain J.: On Melnikov’s persistency problem. Math. Res. Lett. 4, 445–458 (1997)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematics Studies 158, Princeton, NJ: Princeton University Press, 2005

  12. Chierchia L.: A direct method for the construction of solutions of the Hamilton-Jacobi equation. Meccanica 25, 246–252 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chierchia L., Gallavotti G.: Smooth prime integrals for quasi-integrable Hamiltonian systems. Il Nuovo Cimento 67 B, 277–295 (1982)

    ADS  MathSciNet  Google Scholar 

  14. Craig, W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles. Panoramas et Synthèses 9, Paris: Société Math de France, 2000

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Eliasson L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(1), 115–147 (1988)

    MATH  MathSciNet  Google Scholar 

  17. Eliasson L.H., Kuksin S.: KAM for non-linear Schrödinger equation. Ann. of Math. 172, 371–435 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Geng J., Ren X.: Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation. J. Diff. Eq. 249(11), 2796–2821 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gentile G., Procesi M.: Periodic solutions for a class of nonlinear partial differential equations in higher dimension. Commun. Math. Phys. 289(3), 863–906 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Giorgilli A., Morbidelli A.: Superexponential Stability of KAM Tori. J. Stat. Phys. 78(5/6), 1607–1617 (1995)

    ADS  MATH  MathSciNet  Google Scholar 

  21. Jorba A., Villanueva J.: On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems. Nonlinearity 10(4), 783–822 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Kappeler, T., Pöschel, J.: KAM and KdV. Berlin-Heidelberg-New York: Springer, 2003

  23. Kuksin S.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Funkt. Anal. i Pril. 21(3), 22–37, 95 (1987)

    Google Scholar 

  24. Kuksin, S.: Analysis of Hamiltonian PDEs. Oxford Lecture Series in Mathematics and its Applications, 19. Oxford: Oxford University Press, 2000

  25. Kuksin, S., 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 

  26. Pöschel J.: A KAM-Theorem for some nonlinear PDEs. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 23, 119–148 (1996)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Pöschel J.: On the construction of almost periodic solutions for a nonlinear Schrödinger equation. Ergod. Th. and Dyn. Sys. 22, 1537–1549 (2002)

    MATH  Google Scholar 

  29. Rüssmann H.: Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Reg. Chaotic Dyn. 6(2), 119–204 (2001)

    Article  MATH  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi Napoli Federico II, Via Cintia, Monte S. Angelo, 80126, Napoli, Italy

    Massimiliano Berti

  2. Dipartimento di Matematica, Università di Roma 3, Largo San Leonardo Murialdo, 00146, Roma, Italy

    Luca Biasco

Authors
  1. Massimiliano Berti
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  2. Luca Biasco
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Correspondence to Luca Biasco.

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Communicated by G. Gallavotti

Supported by the European Research Council under FP7 “New connections between dynamical systems and Hamiltonian PDEs with small divisors phenomena”.

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Berti, M., Biasco, L. Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs. Commun. Math. Phys. 305, 741–796 (2011). https://doi.org/10.1007/s00220-011-1264-3

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  • DOI: https://doi.org/10.1007/s00220-011-1264-3

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