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

[9972059]西尔伯这个研究项目研究了当液体容器垂直振动时产生的自由表面驻波模式的形成。这个研究得很充分的水动力问题是研究对称破缺参数不稳定性形成图案的一个模型系统。了解实验观察到的“超晶格模式”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具有空间周期性，结构在两个不同的长度尺度上。分析将基于等变分岔理论的方法。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来阐明。该研究项目的一些方面将有助于我们理解弱外部对称破缺对等变分岔问题的影响。本文还将发展一些关于具有时空对称性的周期解的分岔的最新数学结果的应用。最后，将探讨可能的基于对称性的时空模式控制方法。该研究项目旨在对振动液体表面的奇异波型的各种实验观察有一个基本的数学理解。研究结果有望延续到其他技术感兴趣的模式形成系统，如磁流体和非线性光学系统。该研究将有助于我们对振动对多自由度非线性系统影响的基本认识；振动可能是有害的，因为它会导致不稳定，也可能是有益的，因为它会抑制不稳定。该研究还将解决某些流体动力学、非线性光学和化学反应扩散系统中非线性图案形成过程的控制问题。应用数学研究生的培养是研究项目的重要组成部分。

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.