Global Description of Patterns Far from Onset

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NSF Award Abstract - DMS-0202440Mathematical Sciences: Global Description of Patterns Far from OnsetAbstract0202440 NewellNatural patterns turn up all over the place. One sees them on windswept sand dunes, as ripples on long sandy beaches, on plant skins and animal coats, as fingerprints, on optical beams, in cloud formations. The striking similarities between the textures of patterns and their defects which arise in diverse microscopic situations lead one to the conclusion that patterns are macroscopic objects whose behaviors are well described almost everywhere by universal equations that reflect overall symmetry properties shared by the individual systems. For example, the triradius defect on each human fingertip is also seen on optical beams and in magnetic materials. The reason is not that there is any fundamental connection between light and the fetal epidermis but rather that in both contexts the system under stress is two dimensional with translational and rotational invariance. The equations describing the overall pattern will be the same. The central goal of this research is to develop a global description of such stationary patterns far from onset. Up to now, certain individual parts of the textures were well understood, but there was no theory, for example, to say what the total convection pattern would be in an elliptical container whose side walls were heated.This research project will investigate the common mathematical structure that underlies the diverse occurrence of patterns in nature. The work will contribute to the understanding and prediction of important physical phenomena.