Symmetry-Breaking and Pattern Formation, with Applications to Parametrically Excited Surface Waves

这项研究项目致力于研究当液体容器垂直振动时，自由表面驻波图案的形成。这个经过充分研究的流体动力学问题代表了一个通过对称破缺参数不稳定性来研究图案形成的模型系统。了解实验观察到的“超晶格图案”的动力学机制是许多研究的主要焦点。这些二维波的特征是空间周期性的，结构在两个不同的长度尺度上。分析将基于等变分叉理论的方法。范式对称性和空间/时间共振在模式选择过程中的作用将通过分析得到阐明。研究项目的各个方面将有助于我们理解弱外部对称性破缺对等变分叉问题的影响。它还将发展一些最新的关于具有时空对称性的周期解的分支的数学结果的应用。最后，我们将研究可能的基于对称性的时空图样控制方法。该研究项目旨在对振动液体表面奇异波纹的各种实验观测进行基本的数学理解。这一结果有望推广到其他技术感兴趣的图案形成系统，如磁性流体和非线性光学系统。这项研究将有助于我们基本了解振动对多自由度非线性系统的影响；振动可能是有害的，因为它会导致不稳定，或者是有用的，因为它抑制了不稳定。这项研究还将讨论某些流体动力学、非线性光学和化学反应扩散系统中的非线性图案形成过程的控制。应用数学研究生的培训是该研究项目的一个组成部分。

9972059SilberThe research project addresses the formation of free surface standing wave patterns that arise when a container of fluid is vibrated vertically. This well-studied hydrodynamic problem represents a model system for studying pattern formation via symmetry-breaking parametric instability. Understanding the dynamical mechanism for the experimentally observed "superlattice patterns" is a primary focus for much of the research. These two-dimensional wave patterns are characterized as being spatially-periodic, with structure on two disparate length scales. The analysis will be based on methods of equivariant bifurcation theory. The role of normal form symmetries, and spatial/temporal resonance in the pattern selection process will be elucidated by the analysis. Aspects of the research project will contribute to our understanding of the effects of weak external symmetry breaking on equivariant bifurcation problems. It will also develop applications of some recent mathematical results on bifurcation of periodic solutions with spatio-temporal symmetries. Finally, possible symmetry-based methods for controlling spatio-temporal patterns will be investigated.The research project is aimed at a basic mathematical understanding of a variety of experimental observations of exotic wave patterns on the surface of a vibrated liquid. The results are expected to carry over to other pattern-forming systems of technological interest, such as magnetic fluids and nonlinear optical systems. The research will contribute to our basic understanding of the effects of vibration on nonlinear systems with many degrees of freedom; vibration that may be detrimental because it causes instability or useful because it suppresses instability. The research will also address control of the nonlinear pattern-formation process in certain hydrodynamic, nonlinear optical and chemical reaction-diffusion systems. The training of graduate students in applied mathematics is an integral part of the research project.