Patterns in Nature and In the Laboratory

不同微观系统产生的图案纹理的惊人相似性表明，图案是宏观对象，其行为主要依赖于某些整体的大尺度共同对称性，而不是微观细节。三十多年来，这位提出者一直致力于研究模式的宏观特征。其主要思想包括对过多的模式波长取平均值，并引入宏观有序参数，如模式波矢量，与速度和温度等微观变量不同，它在模式波长的距离上变化缓慢。近开始，即应力参数值（如瑞利数、屈曲载荷）接近均匀状态不稳定的阈值时，阶参量是缓慢变化的模式振幅和相位梯度，以及可能存在的某些软（戈德斯通）模式（如低至中等普朗特数对流流体中的平均漂移）。它们满足普适的（纽维尔-怀特黑德-西格尔，复杂的金伯格-兰道）。在远离集合的情况下，幅值服从于相位梯度，而相位梯度和软模态也满足通用偏微分方程（Cross-Newell）。在某些情况下，这些偏微分是梯度流和相位梯度场，模式弛豫是属于调和映射族的泛函的最小值。研究的首要目标是继续开展植物形态和植物叶序性的研究。第二个目标是继续研究表皮脊的形成。第三个目标是进一步深入了解软（Goldstone）模态在弹性板屈曲中的作用。第四个目标是继续我们对远未开始的模式的调查，特别是那些由正则化Cross-Newell （RCN）方程描述的模式。模式无处不在。人们可以在自然界中看到它们，如沙波纹、云结构、指纹、虎纹和豹纹、植物形态和地质构造。在实验室中，它们出现在对流传热实验、铁磁石榴石薄膜实验、大孔径（“脂肪”）激光束实验和火焰前沿实验中。他们都是文学爱好者。现在随便打开一本物理学、化学、生物学、生态学、工程学、数学科学的杂志，你都会发现几篇关于模式形成的文章。产生这种兴趣的原因之一是好奇为什么复杂的微观系统会自己组织成有序的结构。第二个原因是更强大的方法（实验的、数值的、理论的）正变得可用来帮助我们理解模式。第三个原因是，除了科学上的理解之外，还可能有技术上的回报，比如控制强大的激光阵列或信息存储。但也许最纯粹的原因也是最简单的。像所有美丽的事物一样，它们神秘而迷人，充满了开放的挑战，让丰富的想象力去享受和探索。

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.