Patterns in Nature and In the Laboratory

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The striking similarities of pattern textures arising fromdiverse microscopic systems suggest that patterns are macroscopicobjects whose behaviors depend mainly on certain overall large scalecommon symmetries rather than on microscopic details. The proposer hasbeen associated with efforts to exploit the macroscopic character ofpatterns for over thirty years. The main ideas involve averaging overmany pattern wavelengths and introducing macroscopic order parameters,such as the pattern wavevector which, unlike the microscopic variablessuch as velocity and temperature, vary slowly over distances of thepattern wavelength. Near onset, that is for values of a stressparameter (e.g., Rayleigh number, buckling load) close to the thresholdvalue at which the uniform state is unstable, the order parameters arethe slowly varying pattern amplitude and phase gradient and certain soft(Goldstone) modes which ay be present (e.g., mean drift in low tomoderate Prandte number convecting fluids). They satisfy universalpde's (Newell-Whitehead-Segel, complex Gingburg-Landau). Far fromonset, the amplitudes are slaved to the phase gradients and then thephase gradients and the soft modes also satisfy universal pde's(Cross-Newell). In certain cases, these pde's are gradient flows andthe phase gradient fields to which the pattern relaxes are minimizers of functionals which belong to the harmonic map family. The first aim of the proposed research is to continue work on plant patterns and plant phyllotaxis. The second aim is to continue work on epidermal ridge formation. The third aim is to develop further insight into the role of soft (Goldstone) modes in elastic sheet buckling. The fourth aim is to continue our investigations into patterns far from onset and in particular those described by the regularized Cross-Newell (RCN)equation.Patterns are ubiquitous. One sees them in nature as sandripples, as cloud structures, as fingerprints, as tiger stripes andleopard spots, in the morphology of plants, and in geologicalformations. In the laboratory they arise in experiments on heattransport by convection, or ferrimagnetic garnet films, on wide aperture ("fat") laser beams, and on flame fronts. They are all over literature. Open any journal these days in physics, chemistry, biology, ecology, engineering, the mathematical sciences and you will find several articles devoted to pattern formation. One of the reasons for this interest is a curiosity as to why complicated microscopic systemsorganize themselves into ordered structures. A second reason is thatmore powerful methods (experimental, numerical, theoretical) arebecoming available to help us understand patterns. A third reason isthat, in addition to scientific understanding, there may also betechnological payoffs such as control of powerful laser arrays orinformation storage. But perhaps the purest reason is the simplest.Like all beautiful things, they are mysterious and intriguing and fullof open challenges for the fertile imagination to enjoy and explore.