The striking similarities of pattern textures arising from diverse microscopic systems suggest that patterns are macroscopic objects whose behaviors depend mainly on certain overall large scale common symmetries rather than on microscopic details. The proposer has been associated with efforts to exploit the macroscopic character of patterns for over thirty years. The main ideas involve averaging over many pattern wavelengths and introducing macroscopic order parameters, such as the pattern wavevector which, unlike the microscopic variables such as velocity and temperature, vary slowly over distances of the pattern wavelength. Near onset, that is for values of a stress parameter (e.g., Rayleigh number, buckling load) close to the threshold value at which the uniform state is unstable, the order parameters are the slowly varying pattern amplitude and phase gradient and certain soft(Goldstone) modes which ay be present (e.g., mean drift in low to moderate Prandte number convecting fluids). They satisfy universal pde's (Newell-Whitehead-Segel, complex Gingburg-Landau). Far from onset, the amplitudes are slaved to the phase gradients and then the phase gradients and the soft modes also satisfy universal pde's (Cross-Newell). In certain cases, these pde's are gradient flows and the 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 sand ripples, as cloud structures, as fingerprints, as tiger stripes and leopard spots, in the morphology of plants, and in geological formations. In the laboratory they arise in experiments on heat transport 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 systems organize themselves into ordered structures. A second reason is that more powerful methods (experimental, numerical, theoretical) are becoming available to help us understand patterns. A third reason is that, in addition to scientific understanding, there may also be technological payoffs such as control of powerful laser arrays or information storage. But perhaps the purest reason is the simplest. Like all beautiful things, they are mysterious and intriguing and full of open challenges for the fertile imagination to enjoy and explore.
