  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.

Statistical Mechanics;Pattern Formation and Solitons