Advertisement

Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 979–1010 | Cite as

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

  • Shuaishuai Xue
  • Jiansheng Geng
Article
  • 122 Downloads

Abstract

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force
$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$
where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.

Keywords

Schrödinger equation Invariant torus Quasi-periodic solutions KAM theory 

Notes

Acknowledgements

We would like to thank the anonymous referee for many valuable suggestions.

References

  1. 1.
    Bambusi, D.: On long time stability in Hamiltonian perturbations of non-resonant linear PDEs. Nonlinearity 12, 823–850 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berti, M., Bolle, P.: Quasi-periodic solutions with Sobolev regularity of NLS on \(\mathbb{T}^{d}\) and a multiplicative potential. J. Eur. Math. Soc. 15, 229–286 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berti, M., Bolle, P.: Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential. Nonlinearity 25, 2579–2613 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. 148, 363–439 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.: Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. Int. Math. Res. Notices 11, 475–497 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications, Annals of Mathematics Studies 158. Princeton University Press, Princeton (2005)CrossRefGoogle Scholar
  8. 8.
    Bourgain, J.: Nonlinear Schrödinger Equations, Park City Series 5. American Mathematical Society, Providence (1999)Google Scholar
  9. 9.
    Bourgain, J.: On diffusion in high-dimensional Hamiltonian systems and PDE. J. Anal. Math. 80, 1–35 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourgain, J., Wang, W.-M.: Quasi-periodic solutions of nonlinear random Schrödinger equations. J. Eur. Math. Soc. 10, 1–45 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211, 498–525 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Commun. Pure Appl. Math. 46, 1409–1498 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Ann. Sc. Norm. Super. Pisa 15, 115–147 (1988)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172, 371–435 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Geng, J., Wu, J.: Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations. J. Math. Phys. 53, 102702 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Geng, J., Xu, X., You, J.: An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Adv. Math. 226, 5361–5402 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Geng, J., You, J.: A KAM Theorem for one dimensional Schrödinger equation with periodic boundary conditions. J. Differ. Equ. 209, 1–56 (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262, 343–372 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Geng, J., You, J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity 19, 2405–2423 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Geng, J., Yi, Y.: Quasi-periodic solutions in a nonlinear Schrödinger equation. J. Differ. Equ. 233, 512–542 (2007)CrossRefzbMATHGoogle Scholar
  21. 21.
    Geng, J., You, J.: A KAM theorem for higher dimensional nonlinear Schrödinger equation. J. Dyn. Differ. Equ. 25, 451–476 (2013)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kappeler, T., Pöschel, J.: KdV & KAM. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21, 192–205 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kuksin, S.B.: Nearly integrable infinite dimensional Hamiltonian systems. In: Lecture Notes in Mathematics, vol. 1556. Springer, Berlin (1993)Google Scholar
  25. 25.
    Kuksin, S.B., Pöschel, J.: Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation. Ann. Math. 143, 149–179 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helvetici 71, 269–296 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pöschel, J.: A KAM theorem for some nonlinear partial differential equations. Ann. Sc. Norm. Super. Pisa 23, 119–148 (1996)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Pöschel, J.: On the construction of almost periodic solutions for a nonlinear Schrödinger equations. Ergod. Theory Dyn. Syst. 22, 1537–1549 (2002)CrossRefzbMATHGoogle Scholar
  29. 29.
    Procesi, M., Procesi, C.: A normal form for the Schrödinger equation with analytic nonlinearities. Commun. Math. Phys. 312, 501–557 (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Procesi, C., Procesi, M.: A KAM algorithm for the resonant nonlinear Schrödinger equation. Adv. Math. 272, 399–470 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Procesi, M., Xu, X.: Quasi–Töplitz functions in KAM theorem. SIAM J. Math. Anal. 45, 2148–2181 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang, W.M.: Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions. Duke Math. J. 165, 1129–1192 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Wayne, C.E.: Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory. Commun. Math. Phys. 127, 479–528 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Xu, J., Qiu, Q., You, J.: A KAM theorem of degenerate infinite dimensional Hamiltonian systems (I). Sci. China Ser. A Math. 39, 372–383 (1996)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Xu, J., Qiu, Q., You, J.: A KAM theorem of degenerate infinite dimensional Hamiltonian systems (II). Sci. China Ser. A Math. 39, 384–394 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Differ. Equ. 230, 213–274 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

Personalised recommendations