Mathematical Sciences: Evolution of Structures in Nonlinear PDE's

[404374] Muraki提议的工作主题集中在非线性模型中空间结构的演变，这些模型来源于当前感兴趣的物理系统：a)向列液晶中的光自聚焦，b)化学反应中的界面图案形成，c)复杂流体中的流动结构。特别是，数学目标是建立与简单非线性系统中弱湍流行为发展有关的相干结构、不稳定性和动力学之间的关系。基本策略包括主要结构模态的识别，以及它们的动力学和相互作用特性的确定。采用渐近分析、动力系统方法和数值计算相结合的方法，清晰地说明了空间结构由非线性产生和维持的机制。这反过来又提供了一个分析基础，从而导致对潜在非线性现象的物理理解。一个熟悉的时空演变的例子是经常在晚间新闻中播放的卫星天气图的视频记录。虽然空间特征随时间演变的图像是许多物理和生物现象的共同主题，但我们对这些过程的定量理解实际上是相当有限的。这项提议的研究调查了异常复杂模式的模型方程——光学激光束的破裂，以及表面反应中化学物质的扩散——这些模式最近才在实验室中产生，但在设计和控制新的精细尺度过程方面有着未来的希望。

9404374 Muraki The theme of proposed work centers on the evolution of spatial structure in nonlinear models which derive from physical systems of current interest: a) self-focussing of light in nematic liquid crystals, b) interfacial pattern formation in chemical reactions, c) flow structures in complex fluids. In particular, the mathematical objective is to establish relationships between coherent structures, instability and dynamics as pertains to the development of weakly-turbulent behaviors in simple nonlinear systems. The basic strategy involves both the identification of the primary structural modes, and the determination of their dynamics and interaction properties. A combination of asymptotic analysis, dynamical systems methods and numerical computation are employed to obtain clear illustrations of the mechanisms by which spatial structures are created and sustained by nonlinearity. This, in turn, provides an analytical basis leading to a physical understanding for the underlying nonlinear phenomena. A familiar example of a spatio-temporal evolution is the video record of satellite weather maps as often shown on the evening news. While the imagery of spatial features that evolve over time is a common theme for many physical and biological phenomena, our quantitative understanding of these processes is actually quite limited. The proposed research investigates the model equations for unusually complex patterns -- the break-up of optical laser beams, and the spreading of chemicals in surface reactions -- which have only recently been produced in the laboratory, but hold future promise for the design and control of new fine-scale processes.