Phase diffusion and phase propagation: interesting connections

We explore parallels between Whitham theory (nonlinear geometrical optics) applied to gradient systems, such as high Prandtl number pattern forming convection layers, and Hamiltonian systems, such as superfluids at zero temperature and thin elastic shells. In particular, we discuss certain universal features such as the canonical nature of the averaged equations and the relation between the conditions for the onset of vortices, the Eckhaus and Landau criteria respectively. We also show that there is an analogue to the zig-zag instability experienced by gradient systems for Hamiltonian systems and discuss how the asymptotic states may relate. We discuss a new approach for obtaining weak and singular solutions (concave and convex disclinations in phase gradient systems and their composites, phase grain boundaries) which takes advantage of a geometrical property of the phase surface, namely that it has zero Gaussian curvature almost everywhere. We exploit similar ideas in the Hamiltonian context.