Global Description of Patterns Far from Onset

NSF奖摘要-DMS-0202440数学科学：模式的全球描述远离OnsetAbstract0202440新自然模式随处可见。人们可以在狂风肆虐的沙丘上看到它们，在长长的沙滩上看到涟漪，在植物皮肤和动物皮毛上看到它们，作为指纹，在光束上，在云层中。图案的纹理与其在不同微观情况下出现的缺陷之间的惊人相似之处导致人们得出结论：图案是宏观对象，其行为几乎在任何地方都可以用反映单个系统共享的整体对称属性的普遍方程来很好地描述。例如，每个人指尖上的三半径缺陷也可以在光束和磁性材料中看到。原因不是因为光和胎儿的表皮之间有任何根本的联系，而是在这两种情况下，压力下的系统都是二维的，具有平移和旋转不变性。描述整体模式的方程式将是相同的。这项研究的中心目标是对这种远离发病的平稳模式进行全球描述。到目前为止，人们已经很好地了解了纹理的某些个别部分，但还没有理论来说明侧壁被加热的椭圆形容器中的总对流图案会是什么。这项研究项目将调查自然界中图案不同出现的共同数学结构。这项工作将有助于理解和预测重要的物理现象。

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.