Towards a universal theory for natural patterns

Abstract

Our goal is to find a macroscopic description of patterns that both unifies and simplifies classes of externally stressed, dissipative, pattern forming systems, such as convecting fluids, liquid crystals, wideband lasers, that are seemingly unrelated at the microscopic level. We construct an order parameter equation which provides a controlled approximation of the original microscopic field in the limit of large aspect ratios. It is built from, and is a regularization of, the Cross-Newell phase diffusion equation obtained by averaging over the local periodicity of the pattern. Unlike the latter, it is valid for all wavenumbers and can correctly capture the nucleation, shape and nontrivial properties of the far fields of disclinations, dislocations and grain boundaries. It reduces to the Cross-Newell equation away from pattern singularities and to the Newell-Whitehead-Segel equation near onset. As a consequence, it correctly determines all the long wave instability boundaries (zig-zag, Eckhaus-skew-varicose) of the Busse balloon. Far from onset, the order parameter is a real variable but its equation involves a functional corresponding to its local amplitude. The local amplitude and phase, required for the order parameter equation and the reconstruction of the approximation to the original field respectively, are extracted from the order parameter field by wavelet analysis. Numerical comparisons between solutions of the original equation and the regularized equation are carried out. We also explore a new class of singular and weak solutions of the Cross-Newell equation which take account of the energetics of defects as well as their topologies. These solutions correspond to convex and concave disclinations and their composites, including saddles, vortices, targets, dislocations and two new objects, handles and bridges. Finally, we show that phase grain boundaries, lines across which the wavevector is discontinuous but the phase is continuous are captured by shock solutions of the phase diffusion equation.