Statistical Mechanics of the Nonlinear Schr6dinger Equation

is unbounded below and the system will, under certain conditions, develop (selffocusing) singularities in a finite time. We show that, when s is the circle and the L 2 norm of the field (which is conserved by the dynamics) is bounded by N, the Gibbs measure v 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.