Statistical mechanics of the nonlinear Schrödinger equation. (English) Zbl 1084.82506

Summary: We investigate the statistical mechanics of a complex field whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves in a plasma, a propagating laser field in a nonlinear medium, and other phenomena. Their Hamiltonian
\[ H(\phi) = \int_{\Omega}[{1\over 2}|\nabla\phi|^2 - (1/p)|\phi|^p]\text{d}x \]
is unbounded below and the system will, under certain conditions, develop (self-focusing) singularities in a finite time. We show that, when \(\Omega\) is the circle and the \(L^2\) norm of the field (which is conserved by the dynamics) is bounded by \(N\), the Gibbs measure \(\nu\) obtained is absolutely continuous with respect to Wiener measure and normalizable if and only if \(p\) and \(N\) are such that classical solutions exist for all time—no collapse of the solitons. This measure is essentially the same as that of a one-dimensional version of the more realisitc Zakharov model of coupled Langmuir and ion acoustic waves in a plasma. We also obtain some properties of the Gibbs state, by both analytic and numerical methods, as \(N\) and the temperature are varied.


82B05 Classical equilibrium statistical mechanics (general)
35Q55 NLS equations (nonlinear Schrödinger equations)
82D10 Statistical mechanics of plasmas
Full Text: DOI


[1] J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617-656 (1985). · Zbl 0989.37516
[2] V. E. Zakharov,Sov. Phys. JETP 35:908-914 (1972).
[3] M. V. Goldman,Rev. Mod. Phys. 56:709-735 (1984).
[4] H. A. Rose, D. F. DuBois, and B. Bezzerides,Phys. Rev. Lett. 58:2547-2550 (1985).
[5] D. W. McLaughlin, G. C. Papanicolaou, C. Sulem, and P. L. Sulem,Phys. Rev. A 34:1200-1210 (1986).
[6] D. Ruelle,Statistical Mechanics (Benjamin, Reading, Massachusetts, 1974).
[7] J. Glimm and A. Jaffe,Quantum Physics (Springer-Verlag, New York, 1981). · Zbl 0461.46051
[8] J. Ginibre and G. Velo,Ann. Inst. Henri Poincaré 28:287-316 (1978).
[9] V. E. Zakharov and A. B. Shabat,Sov. Phys. JETP 34:62-69 (1972).
[10] D. Russell, D. F. DuBois, and H. A. Rose,Phys. Rev. Lett. 56:838-841 (1986).
[11] M. Weinstein,Commun. Math. Phys. 87:567-576 (1983). · Zbl 0527.35023
[12] S. H. Schochet and M. I. Weinstein,Commun. Math. Phys. 106:569-580 (1986). · Zbl 0639.76054
[13] H. Tasso,Phys. Lett. A 120:464-465 (1987).
[14] G. Pelletier,J. Plasma Phys. 24:287-297 (1980).
[15] G. Pelletier,J. Plasma Phys. 24:421-443 (1980).
[16] G. Z. Sun, D. R. Nicholson, and H. A. Rose,Phys. Fluids 28:2395-2405 (1985). · Zbl 0583.76065
[17] R. H. Kraichnan and D. Montgomery,Rep. Prog. Phys. 43:547-619 (1979).
[18] L. Nirenberg,Commun. Pure Appl. Math. 8:648-674 (1955). · Zbl 0067.07602
[19] R. T. Glassey,J. Math. Phys. 18:1794-1797 (1977). · Zbl 0372.35009
[20] O. Kavian, A Remark on the blowing-up of solutions to the Cauchy problem for the non-linear Schrödinger equation, preprint, Lab. Ana. Nu., Universite P. & M. Curie. · Zbl 0638.35043
[21] B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979). · Zbl 0434.28013
[22] W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York, 1971). · Zbl 0219.60003
[23] D. Dürr and A. Bach,Commun. Math. Phys. 60:153-170 (1978). · Zbl 0377.60044
[24] G. H. Hardy and J. E. Littlewood,Acta Mathematica 37:193-238 (1914).
[25] E. Hille and R. S. Phillips,Functional Analysis and Semi-Groups (American Mathematical Society, Providence, Rhode Island, 1957). · Zbl 0078.10004
[26] A. Zygmund,Trigonometrical Series (Warsaw, 1935). · Zbl 0011.01703
[27] B. Simon,The P(?) Euclidean (Quantum) Field Theory (Princeton University Press, Princeton, New Jersey, 1974). · Zbl 1175.81146
[28] G. Papanicolaou, private communication.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.