Advertisement

Wave Turbulence and Complete Integrability

  • Patrick GérardEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 83)

Abstract

The most famous completely integrable PDEs—KdV and 1D cubic NLS—do not display any kind of wave turbulence, since the conservation laws control high regularity, making impossible the long time appearance of small scales. In this course, I will discuss a new type of integrable infinite-dimensional system, posed on functions of one variable, allowing dramatic growth of high Sobolev norms. The analysis is connected to the solution of an inverse spectral problem for Hankel operators from classical analysis. I will also discuss how this phenomenon can be exported to some Hamiltonian PDEs which can be seen as perturbations of this new type of integrable system.

References

  1. 1.
    J. Bourgain. Problems in Hamiltonian PDE’s. Geom. Funct. Anal., (Special Volume, Part I):32–56, 2000. GAFA 2000 (Tel Aviv, 1999).Google Scholar
  2. 2.
    P. Gérard and S. Grellier. The cubic Szegő equation. Ann. Sci. Éc. Norm. Supér. (4), 43(5):761–810, 2010.MathSciNetCrossRefGoogle Scholar
  3. 3.
    P. Gérard and S. Grellier. Effective integrable dynamics for a certain nonlinear wave equation. Anal. PDE, 5(5):1139–1155, 2012.MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Gérard and S. Grellier. Invariant tori for the cubic Szegö equation. Inventiones mathematicae, 187(3):707–754, 2012.MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Gérard and S. Grellier. An explicit formula for the cubic Szegő equation. Transactions of the American Mathematical Society, 367(4):2979–2995, 2015.MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Gérard and S. Grellier. The cubic Szegő equation and Hankel operators, volume 389 of Astérisque. Société mathématique de France, 2017.Google Scholar
  7. 7.
    P. Gérard, E. Lenzmann, O. Pocovnicu, and P. Raphaël. A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line. Ann. PDE 4(1), Art. 7, 2018.Google Scholar
  8. 8.
    Z. Hani. Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations. Arch. Ration. Mech. Anal., 211(3):929–964, 2014.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Z. Hani, B. Pausader, N. Tzvetkov, and N. Visciglia. Almost sure global well-posedness for fractional cubic Schrödinger equation on torus. Forum Math. Pi3, 2015.Google Scholar
  10. 10.
    J. Krieger, E. Lenzmann, and P. Raphaël. Nondispersive solutions to the l2-critical half- wave equation. Arch. Ration. Mech. Anal., 209(1):61–129, 2013.MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 21:467–490, 1968.MathSciNetCrossRefGoogle Scholar
  12. 12.
    A Majda, D. McLaughlin, and E. Tabak. A one dimensional model for dispersive wave turbulence. Nonlinear Science, 6:9–44, 1997.MathSciNetCrossRefGoogle Scholar
  13. 13.
    V. Peller. Hankel operators and their applications. Springer Science & Business Media, 2012.Google Scholar
  14. 14.
    O. Pocovnicu. Explicit formula for the solution of the szegő equation on the real line and applications. Discrete Contin. Dyn. Syst. A, 31(3):607–649, 2011.MathSciNetCrossRefGoogle Scholar
  15. 15.
    O. Pocovnicu. Traveling waves for the cubic szegő equation on the real line. Anal. PDE, 4(3):379–404, 2011.MathSciNetCrossRefGoogle Scholar
  16. 16.
    J. Thirouin. On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle. Annales de l’Institut Henri Poincaré (C), Nonlinear Analysis, 34(509–531), 2016.MathSciNetCrossRefGoogle Scholar
  17. 17.
    H. Xu. Large time blowup for a perturbation of the cubic Szegő equation. Analysis & PDE, 7(3):717–731, 2014.MathSciNetCrossRefGoogle Scholar
  18. 18.
    H. Xu. The cubic Szegő equation with a linear perturbation. Preprint, arXiv:1508.01500, August 2015.Google Scholar
  19. 19.
    H. Xu. Unbounded sobolev trajectories and modified scattering theory for a wave guide nonlinear schrödinger equation. Mathematische Zeitschrift, 286(1–2):443–489, 2017.MathSciNetCrossRefGoogle Scholar
  20. 20.
    V. E. Zakharov and A. B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Ž. Èksper. Teoret. Fiz., 61(1):118–134, 1971.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance

Personalised recommendations