Brydges, D.C., Slade, G.
Statistical mechanics of the 2-dimensional
focussing nonlinear Schroedinger equation
(65K, LaTeX)
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.