Mathematische Annalen

, Volume 359, Issue 1–2, pp 471–536 | Cite as

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

  • Pietro Baldi
  • Massimiliano BertiEmail author
  • Riccardo Montalto


We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash–Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients.

Mathematics Subject Classification (2010)

37K55 35Q53 



We warmly thank W. Craig for many discussions about the reduction approach of the linearized operators and the reversible structure, and P. Bolle for deep observations about the Hamiltonian case. We also thank T. Kappeler, M. Procesi for many useful comments.


  1. 1.
    Baldi, P.: Periodic solutions of forced Kirchhoff equations. Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (5) 8, 117–141 (2009)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Baldi, P.: Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type. Ann. I. H. Poincaré (C) Anal. Non Linéaire 30(1), 33–77 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baldi, P., Berti, M., Montalto, R.: A note on KAM theory for quasi-linear and fully nonlinear KdV. Rend. Lincei Mat. Appl. 24, 437–450 (2013)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bambusi, D., Graffi, S.: Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods. Commun. Math. Phys. 219, 465–480 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Berti, M., Biasco, L.: Branching of Cantor manifolds of elliptic tori and applications to PDEs. Commun. Math. Phys 305(3), 741–796 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Berti, M., Biasco, P., Procesi, M.: KAM theory for the Hamiltonian DNLW. Ann. Sci. Éc. Norm. Supér. (4), t. 46 (2013), fascicule 2, 301–272Google Scholar
  7. 7.
    Berti, M., Biasco, L., Procesi, M.: KAM for the reversible derivative wave equations. Arch. Ration. Mech. Anal (to appear)Google Scholar
  8. 8.
    Berti, M., Bolle, P.: Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math. J. 134, 359–419 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Berti, M., Bolle, P.: Quasi-periodic solutions with Sobolev regularity of NLS on \(\mathbb{T}^d\) with a multiplicative potential. Eur. J. Math. 15, 229–286 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Berti, M., Bolle, P.: Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Berti, M., Bolle, P., Procesi, M.: An abstract Nash–Moser theorem with parameters and applications to PDEs. Ann. I. H. Poincaré 27, 377–399 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Int. Math. Res. Notices, no. 11 (1994)Google Scholar
  13. 13.
    Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of \(2D\) linear Schrödinger equations. Ann. Math. 148, 363–439 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Bourgain, J.: Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations. Chicago Lectures in Math., pp. 69–97. Univ. Chicago Press, Chicago (1999)Google Scholar
  15. 15.
    Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. In: Annals of Mathematics Studies, vol. 158. Princeton University Press, Princeton (2005)Google Scholar
  16. 16.
    Craig, W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles. In: Panoramas et Synthèses, vol. 9. Société Mathématique de France, Paris (2000)Google Scholar
  17. 17.
    Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equation. Commun. Pure Appl. Math. 46, 1409–1498 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Delort, J.-M.: A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein–Gordon equation on \(\mathbb{S}^1\). Astérisque 341 (2012)Google Scholar
  19. 19.
    Eliasson, L.H., Kuksin, S.: KAM for non-linear Schrödinger equation. Ann. Math. 172, 371–435 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Eliasson, L.H., Kuksin, S.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286, 125–135 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Geng, J., Xu, X., You, J.: An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Adv. Math. 226(6), 5361–5402 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Grebert, B., Thomann, L.: KAM for the quantum harmonic oscillator. Commun. Math. Phys. 307(2), 383–427 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Iooss, G., Plotnikov, P.I., Toland, J.F.: Standing waves on an infinitely deep perfect fluid under gravity. Arch. Rational Mech. Anal. 177(3), 367–478 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Iooss, G., Plotnikov, P.I.: Small divisor problem in the theory of three-dimensional water gravity waves. Mem. Am. Math. Soc. 200, no. 940 (2009)Google Scholar
  26. 26.
    Iooss, G., Plotnikov, P.I.: Asymmetrical three-dimensional travelling gravity waves. Arch. Rational Mech. Anal. 200(3), 789–880 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kappeler, T., Pöschel, J.: KAM and KdV. Springer, Berlin (2003)CrossRefGoogle Scholar
  28. 28.
    Klainermann, S., Majda, A.: Formation of singularities for wave equations including the nonlinear vibrating string. Commun. Pure Appl. Math. 33, 241–263 (1980)CrossRefGoogle Scholar
  29. 29.
    Kuksin, S.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen. 21(3), 22–37, 95 (1987)Google Scholar
  30. 30.
    Kuksin, S.: A KAM theorem for equations of the Korteweg–de Vries type. Rev. Math. Math Phys. 10(3), 1–64 (1998)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Kuksin, S.: Analysis of Hamiltonian PDEs. In: Oxford Lecture Series in Mathematics and its Applications, vol. 19. Oxford University Press, Oxford (2000)Google Scholar
  32. 32.
    Kuksin, S., Pöschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. (2) 143, 149–179 (1996)CrossRefzbMATHGoogle Scholar
  33. 33.
    Lax, P.: Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys. 5, 611–613 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Liu, J., Yuan, X.: Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient. Commun. Pure Appl. Math. 63(9), 1145–1172 (2010)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Liu, J., Yuan, X.: A KAM theorem for hamiltonian partial differential equations with unbounded perturbations. Commun. Math. Phys. 307, 629–673 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Moser, J.: A rapidly convergent iteration method and non-linear partial differential equations-I. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (3) 20(2), 265–315 (1966)zbMATHGoogle Scholar
  37. 37.
    Pöschel, J.: A KAM-theorem for some nonlinear partial differential equations. Ann. Sci. Norm. Sup. Pisa Cl. Sci. (4) 23, 119–148 (1996)zbMATHGoogle Scholar
  38. 38.
    Procesi, C., Procesi, M.: A KAM algorithm for the completely resonant nonlinear Schrödinger equation (2012, preprint)Google Scholar
  39. 39.
    Procesi, M., Xu, X.: Quasi-Töplitz Functions in KAM Theorem. SIAM J. Math. Anal. 45(4), 2148–2181 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Rabinowitz, P.H.: Periodic solutions of nonlinear hyperbolic partial differential equations, Part I. Commun. Pure Appl. Math. 20, 145–205 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Rabinowitz, P.H.: Periodic solutions of nonlinear hyperbolic partial differential equations. Part II. Commun. Pure Appl. Math. 22, 15–39 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Wang, W.M.: Supercritical nonlinear Schrödinger equations I: quasi-periodic solutions (2010, preprint)Google Scholar
  43. 43.
    Wayne, E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Zhang, J., Gao, M., Yuan, X.: KAM tori for reversible partial differential equations. Nonlinearity 24, 1189–1228 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Pietro Baldi
    • 1
  • Massimiliano Berti
    • 1
    • 2
    Email author
  • Riccardo Montalto
    • 2
  1. 1.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università degli Studi Napoli Federico IINaplesItaly
  2. 2.SISSATriesteItaly

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