Patterns, defects and integrability

In this paper, recent results on the behavior of roll patterns in a class o f problems typified by high Prandtl number convection are presented. A key finding is that the Gaussian curvature of the "crumpled" phase surface. whi ch 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 con siderable mathematical simplification in that the fourth order nonlinear di ffusion equation governing stationary states can be effectively reduced to the linear Helmholtz equation. The observed patterns have much is common wi th the deformation of thin elastic sheets. Copyright (C) 1998 Published by Elsevier Science B.V.