Landau-Ginzburg equations with random initial conditions

Abstract

The long-time behaviour of solutions of the cubic reaction-diffusion Landau-Ginzburg equation submitted to random homogeneous initial conditions and the linear heat equation with Gaussian random potential are shown to be equivalent. This entails that intermittent patterns as well as a long-normal multifractal structure emerge from the temporal evolution. These fluctuations are arranged in a hierarchical fashion reminiscent of a spin-glass transition and of the phase of Gaussian Kolmogorov-Mandelbrot cascades. Extensions to related equations where the cubic term is replaced by other monomials are also derived, they correspond to the Levy law generalization of the log-normal probability distribution.