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A Normal Form for the Schrödinger Equation with Analytic Non-linearities

Communications in Mathematical Physics volume 312, pages 501–557 (2012)Cite this article

Abstract

We discuss a class of normal forms of the completely resonant non-linear Schrödinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study.

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References

  1. Bambusi D., Grébert B. (2006) Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J. 135(3): 507–567

    MathSciNet  MATH  Article  Google Scholar 

  2. Berti, M., Bolle, Ph.: Quasi-periodic solutions Sobolev regularity of NLS on \({ \mathbb{T}^d}\) with a multiplicative potential. to appear on Eur. Jour. Math., http://arxiv.org/abs/1012.1427v1 [math.Ap], 2010

  3. Bombieri E., Pila J. (1989) The number of integral points on arcs and ovals. Duke Math. J. 59(2): 337–357

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

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

  6. Chavaudret C. (2011) Reducibility of quasiperiodic cocycles in linear Lie groups. Erg. The. Dyn. Sys. 31(3): 741–769

    MathSciNet  MATH  Article  Google Scholar 

  7. Colliander J., Keel M., Staffilani G., Takaoka H., Tao T. (2010) Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math. 181(1): 39–113

    MathSciNet  ADS  MATH  Article  Google Scholar 

  8. Craig W., Wayne C.E. (1993) Newton’s method and periodic solutions of nonlinear wave equations. Comm. Pure Appl. Math. 46(11): 1409–1498

    MathSciNet  MATH  Article  Google Scholar 

  9. Eliasson L.H. (2001) Almost reducibility of linear quasi-periodic systems. Proc. Symp. Pure Math. 69: 679–705

    MathSciNet  Google Scholar 

  10. Geng J., Yi Y. (2007) Quasi-periodic solutions in a nonlinear Schrüdinger equation. J. Diff. Eqs. 233: 512–542

    MathSciNet  MATH  Article  Google Scholar 

  11. Geng J., You J., Xu X. (2011) An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation. Adv. Math. 226(6): 5361–5402

    MathSciNet  MATH  Article  Google Scholar 

  12. Gentile G., Procesi M. (2008) Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension. J. Diff. Eqs. 245(11): 3253–3326

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  ADS  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  15. Pöschel J. (1996) A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(1): 119–148

    MathSciNet  MATH  Google Scholar 

  16. Procesi M. (2010) A normal form for beam and non-local nonlinear Schrödinger equations. J. of Physics A: Math. Theor. 43(43): 434028

    MathSciNet  ADS  Article  Google Scholar 

  17. Wang, W.M.: Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities. Preprint, http://arxiv.org/abs/0907.3409v2 [math.Ap], 2009

  18. Wang, W.M.: Supercritical nonlinear Schrödinger equations i: Quasi-periodic solutions. Preprint, http://arxiv.org/abs/1007.0156v1 [math.Ap], 2010

  19. Wang, W.M.: Supercritical nonlinear Schrödinger equations ii: Almost global existence. Preprint, http://arxiv.org/abs/1007.0154v1 [math.Ap], 2010

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Authors and Affiliations

  1. Department of Mathematics, University of Naples Federico II, Naples, 80138, Italy

    M. Procesi

  2. Department of Mathematics, University of Rome, La Sapienza, Rome, 00185, Italy

    C. Procesi

Authors
  1. M. Procesi
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  2. C. Procesi
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Corresponding author

Correspondence to C. Procesi.

Additional information

Supported by ERC grant “New connections between dynamical systems and Hamiltonian PDEs” and partially by the PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”.

Communicated by G. Gallavotti

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Procesi, M., Procesi, C. A Normal Form for the Schrödinger Equation with Analytic Non-linearities. Commun. Math. Phys. 312, 501–557 (2012). https://doi.org/10.1007/s00220-012-1483-2

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  • DOI: https://doi.org/10.1007/s00220-012-1483-2

Keywords

  • Normal Form
  • Cayley Graph
  • Geometric Realization
  • Geometric Graph
  • Black Vertex