Communications in Mathematical Physics

, Volume 244, Issue 1, pp 187–208 | Cite as

A Statistical Approach to the Asymptotic Behavior of a Class of Generalized Nonlinear Schrödinger Equations

  • Richard S. Ellis
  • Richard Jordan
  • Peter Otto
  • Bruce Turkington
Article

Abstract

A statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear Schrödinger-type which govern non-integrable, non-singular dynamics. In a bounded domain the solutions of these equations have been shown numerically to tend in the long-time limit toward a Gibbsian statistical equilibrium state consisting of a ground-state solitary wave on the large scales and Gaussian fluctuations on the small scales. The main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit. The large deviation principle pertains to a process governed by a Gibbs ensemble that is canonical in energy and microcanonical in particle number. Some supporting Monte-Carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation, especially the high frequency growth of the dispersion relation. The large deviation principle for the process governed by the Gibbs ensemble is based on a large deviation principle for Gaussian processes, for which two independent proofs are given.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bidegaray,~B.: Invariant measures for some partial differential equations. Physica D 82, 340–364 (1995)MathSciNetMATHGoogle Scholar
  2. 2.
    Binder,~K., Heermann,~D.W.: Monte Carlo Simulation in Statistical Physics. Fourth edition. Springer Series in Solid-State Sciences, Vol. 80, Berlin: Springer-Verlag, 2002Google Scholar
  3. 3.
    Birkhoff,~G., Rota,~G.-C.: Ordinary Differential Equations. Second edition. Waltham: Blaisdell Publishing Co., 1969Google Scholar
  4. 4.
    Biskamp,~D.: Nonlinear Magnetohydrodynamics. Cambridge Monographs in Plasma Physics. Cambridge: Cambridge Univ. Press,1993Google Scholar
  5. 5.
    Bolthausen,~E.: On the probability of large deviations in Banach spaces. Ann. Probab. 12, 427–435 (1984)MathSciNetMATHGoogle Scholar
  6. 6.
    Boucher,~C., Ellis,~R.S., Turkington,~B.: Derivation of maximum entropy principles in two-dimensional turbulence via large deviations. J. Stat. Phys 98, 1235–1278 (2000)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Bourgain,~J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166, 1–26 (1994)MathSciNetMATHGoogle Scholar
  8. 8.
    Bouchet,~F., Sommeria,~J.: Emergence of intense jets and Jupiter’s Great Red Spot as maximum-entropy structures. J. Fluid Mech. 464, 165–207 (2002)CrossRefMATHGoogle Scholar
  9. 9.
    Cai,~D., Majda,~A.J., McLaughlin,~D.W., Tabak,~E.G.: Spectral bifurcations in dispersive wave turbulence. Proc. Nat. Acad. Sci. 96, 14216–14221 (1999)CrossRefMATHGoogle Scholar
  10. 10.
    Cai,~D., McLaughlin,~D.W.: Chaotic and turbulent behavior of unstable 1D nonlinear dispersive waves. J. Math. Phys. 41, 4125–4153 (2000)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Dembo,~A., Zeitouni,~O.: Large Deviations Techniques and Applications. Second edition. New York: Spring-Verlag, 1998Google Scholar
  12. 12.
    DiBattista,~M.T., Majda,~A.J., Grote,~M.J.: Meta-stability of equilibrium statistical structures for prototype geophysical flows with damping and driving. Physica D 151, 271–304 (2001)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Dowling,~T.E.: Dynamics of Jovian atmospheres. Ann. Rev. Fluid Mech. 27, 293–334 (1995)CrossRefGoogle Scholar
  14. 14.
    Dupuis,~P., Ellis,~R.S.: A Weak Convergence Approach to the Theory of Large Deviations. New York: John Wiley & Sons, 1997Google Scholar
  15. 15.
    Dyachenko,~S., Zakharov,~V.E., Pushkarev,~A.N., Shvets,~V.F., Yan’kov,~V.V.: Soliton turbulence in nonintegrable wave systems. Soviet Phys. JETP 69, 1144–1147 (1989)Google Scholar
  16. 16.
    Ellis,~R.S.: Entropy, Large Deviations and Statistical Mechanics. New York: Springer-Verlag, 1985Google Scholar
  17. 17.
    Ellis,~R.S., Haven,~K., Turkington,~B.: Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101, 999–1064 (2000)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Gikhman,~I.I., Skorohod,~A.V.: The Theory of Stochastic Processes I. Trans. by S. Kotz, Berlin: Springer-Verlag, 1974Google Scholar
  19. 19.
    Hasegawa,~A.: Self-organization processes in continuous media. Adv. Phys. 34, 1–42 (1985)MathSciNetMATHGoogle Scholar
  20. 20.
    Isichenko,~M.B., Gruzinov,~A.V.: Isotopological relaxation, coherent structures, and Gaussian turbulence in two-dimensional magnetohydrodynamics. Phys. Plasmas 1, 1802–1816 (1994)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Itô,~K., McKean,~H.P.: Diffusion Processes and Their Sample Paths. New York/Berlin: Academic Press/Springer Verlag, 1965Google Scholar
  22. 22.
    Jordan,~R., Josserand,~C.: Self-organization in nonlinear wave turbulence. Phys. Rev. E 61, 1527–1539 (2000)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Jordan,~R., Josserand,~C.: Statistical equilibrium states for the nonlinear Schrödinger equation. Math. Comp. Simulation 55, 433–447 (2001)CrossRefMATHGoogle Scholar
  24. 24.
    Jordan,~R., Turkington,~B.: Ideal magnetofluid turbulence in two dimensions J. Stat. Phys. 87, 661–695 (1997)MathSciNetMATHGoogle Scholar
  25. 25.
    Jordan,~R., Turkington,~B., Zirbel,~C.L.: A mean-field statistical theory for the nonlinear Schrödinger equation. Physica D 137, 353–378 (2000)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Kevrekidis,~P.G., Rasmussen,~K.O., Bishop,~A.R.: The discrete nonlinear Schrödinger equation: A survey of recent results. Int. J. Mod. Phys. B. 15, 2833–2900 (2001)CrossRefGoogle Scholar
  27. 27.
    Lebowitz,~J.L., Rose,~H.A., Speer,~E.R.: Statistical mechanics of a nonlinear Schrödinger equation. J. Stat. Phys., 50, 657–687 (1988)Google Scholar
  28. 28.
    Majda,~A.J., McLaughlin,~D.W., Tabak,~E.G.: A one-dimensional model for dispersive wave turbulence. J. Nonlinear Sci. 7, 9–44 (1997)MathSciNetMATHGoogle Scholar
  29. 29.
    Marcus,~P.S.: Jupiter’s Great Red Spot and other vortices. Annual Rev. Astronomy and Astrophys. 31, 523–573 (1993)CrossRefGoogle Scholar
  30. 30.
    McKean,~H.P.: Statistical mechanics of nonlinear wave equations IV. Cubic Schrödinger. Commun. Math. Phys.168, 479–491 (1995)Google Scholar
  31. 31.
    Rasmussen,~J.J., Rypdal,~K.: Blow-up in nonlinear Schroedinger equations–I: A general review. Physica Scripta 33, 481–504 (1986)MathSciNetMATHGoogle Scholar
  32. 32.
    Segre,~E., Kida,~S.: Late states of incompressible 2d decaying vorticity fields. Fluid Dyn. Res. 23, 89–112 (1998)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Turkington,~B., Majda,~A.J., Haven,~K., DiBattista,~M.: Statistical equilibrium predictions of jets and spots on Jupiter. Proc. Nat. Acad. Sci. USA 98, 12346–12350 (2001)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Zakharov,~V.E., Pushkarev,~A.N., Shvets,~V.F., Yan’kov,V.V.: Soliton turbulence. JETP Lett. 48, 83–86 (1988)Google Scholar
  35. 35.
    Zhidkov,~P.E.: On an invariant measure for a nonlinear Schrödinger equation. Soviet Math. Dokl. 43, 431–434 (1991)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Richard S. Ellis
    • 1
  • Richard Jordan
    • 2
  • Peter Otto
    • 1
  • Bruce Turkington
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsUSA
  2. 2.Dynamics Technology, Inc.USA

Personalised recommendations