96-136 Brydges, D.C., Slade, G.

Statistical mechanics of the 2-dimensional focussing nonlinear Schroedinger equation (65K, LaTeX) Apr 17, 96

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Abstract. We study a natural construction of an invariant measure for the 2-dimensional periodic focussing nonlinear Schr\"odinger equation, with the critical cubic nonlinearity. We find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased, but that the natural construction, successful for the 1-dimensional case and for the 2-dimensional defocussing case, cannot produce an invariant measure. Our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of Berlin and Kac.

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