TOWARDS A UNIVERSAL THEORY FOR NATURAL PATTERNS

Citation
T. Passot et Ac. Newell, TOWARDS A UNIVERSAL THEORY FOR NATURAL PATTERNS, Physica. D, 74(3-4), 1994, pp. 301-352
Citations number
38
Categorie Soggetti
Mathematical Method, Physical Science",Physics,"Physycs, Mathematical
Journal title
ISSN journal
01672789
Volume
74
Issue
3-4
Year of publication
1994
Pages
301 - 352
Database
ISI
SICI code
0167-2789(1994)74:3-4<301:TAUTFN>2.0.ZU;2-H
Abstract
Our goal is to find a macroscopic description of patterns that both un ifies and simplifies classes of externally stressed, dissipative, patt ern forming systems, such as convecting fluids, liquid crystals, wideb and lasers, that are seemingly unrelated at the microscopic level. We construct an order parameter equation which provides a controlled appr oximation of the original microscopic field in the limit of large aspe ct ratios. It is built from, and is a regularization of, the Cross-New ell phase diffusion equation obtained by averaging over the local peri odicity of the pattern. Unlike the latter, it is valid for all wavenum bers and can correctly capture the nucleation, shape and nontrivial pr operties of the far fields of disclinations, dislocations and grain bo undaries. It reduces to the Cross-Newell equation away from pattern si ngularities and to the Newell-Whitehead-Segel equation near onset. As a consequence, it correctly determines all the long wave instability b oundaries (zig-zag, Eckhaus-skew-varicose) of the Busse balloon. Far f rom onset, the order parameter is a real variable but its equation inv olves a functional corresponding to its local amplitude. The local amp litude and phase, required for the order parameter equation and the re construction of the approximation to the original field respectively, are extracted from the order parameter field by wavelet analysis. Nume rical comparisons between solutions of the original equation and the r egularized equation are carried out. We also explore a new class of si ngular and weak solutions of the Cross-Newell equation which take acco unt of the energetics of defects as well as their topologies. These so lutions correspond to convex and concave disclinations and their compo sites, including saddles, vortices, targets, dislocations and two new objects, handles and bridges. Finally, we show that phase grain bounda ries, lines across which the wavevector is discontinuous but the phase is continuous are captured by shock solutions of the phase diffusion equation.