Statistical mechanics of the 2-dimensional focusing nonlinear Schrodinger equation

We study a natural construction of an invariant measure for the 2-dimensional periodic focusing nonlinear Schrodinger 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 defocusing 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.