The universal behavior of modulated stripe patterns

There are three aims of this paper. The first is to explain the reasons for behavior we had long suspected to be true but the real reasons for which we could never quite nail down. Modulated striped patterns arising from a wide class of gradient microscopic pattern forming systems display universal behavior. Their order parameter evolves according to a phase diffusion equation that derives from an averaged energy that consists of coordinate invariant combinations of the coefficients in the metric and curvature two forms of a well-defined phase surface. The evolution towards universality is asymptotic in that the pattern evolves in such a way that, over longer and longer time scales, many terms from the microscopic energy simply become negligible leaving behind canonical forms common to a wide class of microscopic pattern-forming systems. The second aim is to emphasize with some new results the key role that the Jacobian matrix of the map from physical to order parameter space plays in both two and three dimensions. In two dimensions, it is closely related to the Gaussian curvature of the phase surface. It is a conserved density whose integral over space in two dimensions or on cross-sections in three become boundary terms that measure the topological indices of point defects in two dimensions and loop defects in three. In all dimensions, the Jacobian matrix acting on the order parameter vector, the gradient of the phase, is zero when the local pattern wavenumber is close to its preferred value and this leads to the effective linearization of the phase diffusion equation. The third aim is to honor Hermann Flaschka, a close friend and scientific colleague for over fifty years, an outstanding mathematician, a true gentleman and scholar with an uncanny knack of explaining the most complicated of ideas in the simplest of ways, who passed away last year. Hermann was one of the founding editors of Physica D and served as the coordinating editor for almost twenty years.