Patterns, defects and integrability

In this paper, recent results on the behavior of roll patterns in a class of problems typified by high Prandtl number convection are presented. A key finding is that the Gaussian curvature of the "crumpled" phase surface, which consists of patches with an almost constant wave number, line defects on which most of the free energy is stored and point defects with nontrivial topologies; condenses onto line and point defects. This property allows considerable mathematical simplification in that the fourth order nonlinear diffusion equation governing stationary states can be effectively reduced to the linear Helmholtz equation. The observed patterns have much is common with the deformation of thin elastic sheets.