Inventiones mathematicae

, Volume 183, Issue 1, pp 79–188 | Cite as

Weakly nonlinear Schrödinger equation with random initial data

  • Jani LukkarinenEmail author
  • Herbert Spohn


It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈ℤ d and t∈ℝ. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ t (x) has a limit as λ→0 for t=λ −2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.

Mathematics Subject Classification (2000)

74J20 81Q30 37K60 35Q55 70K70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdesselam, A., Procacci, A., Scoppola, B.: Clustering bounds on n-point correlations for unbounded spin systems. J. Stat. Phys. 136(3), 405–452 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    van Beijeren, H., Lanford, O.E., Lebowitz, J.L., Spohn, H.: Equilibrium time correlation functions in the low-density limit. J. Stat. Phys. 22(2), 237–257 (1980) zbMATHCrossRefGoogle Scholar
  3. 3.
    Benedetto, D., Castella, F., Esposito, R., Pulvirenti, M.: From the N-body Schrödinger equation to the quantum Boltzmann equation: a term-by-term convergence result in the weak coupling regime. Commun. Math. Phys. 277(1), 1–44 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Buttà, P., Caglioti, E., Di Ruzza, S., Marchioro, C.: On the propagation of a perturbation in an anharmonic system. J. Stat. Phys. 127(2), 313–325 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Erdős, L., Salmhofer, M., Yau, H.T.: Quantum diffusion for the Anderson model in the scaling limit. Ann. Henri Poincaré 8(4), 621–685 (2007) CrossRefGoogle Scholar
  6. 6.
    Erdős, L., Salmhofer, M., Yau, H.T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Commun. Math. Phys. 271(1), 1–53 (2007) CrossRefGoogle Scholar
  7. 7.
    Erdős, L., Salmhofer, M., Yau, H.T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit I. The non-recollision diagrams. Acta Math. 200(2), 211–277 (2008) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Erdős, L., Yau, H.T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Commun. Pure Appl. Math. 53(6), 667–735 (2000) CrossRefGoogle Scholar
  9. 9.
    Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn. Springer, New York (1987) Google Scholar
  10. 10.
    Gurevich, V.L.: Transport in Phonon Systems. North-Holland, Amsterdam (1986) Google Scholar
  11. 11.
    Ho, T.G., Landau, L.J.: Fermi gas on a lattice in the van Hove limit. J. Stat. Phys. 87(3), 821–845 (1997) zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Janssen, P.A.E.M.: Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33(4), 863–884 (2003) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Landau, L.J., Luswili, N.J.: Asymptotic expansion of a Bessel function integral using hypergeometric functions. J. Comput. Appl. Math. 132(2), 387–397 (2001) zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Lanford, O.E., Lebowitz, J.L., Lieb, E.H.: Time evolution of infinite anharmonic systems. J. Stat. Phys. 16(6), 453–461 (1977) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of unbounded spins. Commun. Math. Phys. 50(3), 195–218 (1976) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lukkarinen, J., Spohn, H.: Kinetic limit for wave propagation in a random medium. Arch. Ration. Mech. Anal. 183(1), 93–162 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Lukkarinen, J., Spohn, H.: Not to normal order—Notes on the kinetic limit for weakly interacting quantum fluids. J. Stat. Phys. 134(5), 1133–1172 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Lvov, Y.V., Nazarenko, S.: Noisy spectra, long correlations, and intermittency in wave turbulence. Phys. Rev. E 69(6), 066608 (2004) MathSciNetGoogle Scholar
  19. 19.
    Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields: Cluster Expansions. Springer, Dordrecht (1991) zbMATHGoogle Scholar
  20. 20.
    Salmhofer, M.: Clustering of fermionic truncated expectation values via functional integration. J. Stat. Phys. 134(5), 941–952 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124(2–4), 1041–1104 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, Berlin (1999) zbMATHGoogle Scholar
  23. 23.
    Zakharov, V.E., L’Vov, V.S., Falkovich, G.: Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer, Berlin (1992) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsingin yliopistoFinland
  2. 2.Zentrum MathematikTechnische Universität MünchenGarchingGermany

Personalised recommendations