## 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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### References:

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