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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.

MSC:

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