Phase instability and coarsening in two dimensions

Instabilities and pattern formation is the rule in nonequilibrium systems. Selection of a persistent lengthscale, or coarsening (increase of the lengthscale with time) are the two major alternatives. When and under which conditions one dynamics prevails over the other is a longstanding problem, particularly beyond one dimension. It is shown that the challenge can be defied in two dimensions, using the concept of phase diffusion equation. We find that coarsening is related to the \lambda-dependence of a suitable phase diffusion coefficient, D_{11}(\lambda), depending on lattice symmetry and conservation laws. These results are exemplified analytically on prototypical nonlinear equations.