  We consider the nonlinear Schrodinger equation with a logarithmic
nonlinearity in a dispersive regime. We show that the presence of the
nonlinearity affects the large time behavior of the solution: the dispersion is
faster than usual by a logarithmic factor in time and the positive Sobolev
norms of the solution grow logarithmically in time. Moreover, after rescaling
in space by the dispersion rate, the modulus of the solution converges to a
universal Gaussian profile. These properties are suggested by explicit
computations in the case of Gaussian initial data, and remain when an extra
power-like nonlinearity is present in the equation. One of the key steps of the
proof consists in using the Madelung transform to reduce the equation to a
variant of the isothermal compressible Euler equation, whose large time
behavior turns out to be governed by a parabolic equation involving a
Fokker-Planck operator.
