Spatially Localized States in Extended Systems

空间定域结构在流体、非线性光学和催化引起的化学反应等连续系统中很常见。例如，局部对流，光学和化学系统中的斑点，细长结构在压缩下的局部屈曲，沿神经纤维传播的脉冲，振动颗粒介质中的振荡，液滴和流动液膜上的孤立波。这些不同的系统有两个共同点：(i)它们是由均匀强迫驱动的耗散系统，（ii）存在强迫范围，在该范围内应用不同的有限振幅扰动可以导致不同的局部状态。本提案旨在将现有的理论（部分由PI及其同事开发）扩展到更高的维度，并提供对有限域尺寸，各向异性和平移不变性损失对这些结构的起源和性质的影响的全面理解。所使用的技术包括可逆和近可逆系统的分岔理论，以及实际系统的数值分支跟踪和直接数值模拟。许多系统通过产生空间格局来响应空间均匀强迫。这些模式可以采用细胞或斑点的周期性阵列的形式，或者在特定情况下通过产生单个斑点或一组斑点。本建议旨在了解这两种反应之间的关系，并预测后一种反应可能发生的条件。点样响应有许多潜在的应用。在光学中，单个点可以用激光束“写入”和“擦除”，这一过程可用于存储信息。机械结构通常以局部方式屈曲。薄的液体薄膜可能破裂成液滴。使用催化的化学过程可能会退化，因为化学转化在空间中不能均匀进行。这些都是空间局部结构的例子，将作为本项目的一部分进行研究。这些点状结构也可以移动和相互作用。例如，这些移动的结构与沿神经纤维发出的信号有关。该项目旨在预测这些结构的形成并了解它们的基本特性。

Spatially localized structures are common in continuum systems such as fluids, nonlinear optics and the chemical reactions arising in catalysis. Examples are provided by localized convection, spots in optical and chemical systems, localized buckling of slender structures under compression, pulses propagating along neural fibers, oscillons in vibrating granular media, liquid drops, and solitary waves on flowing liquid films. These diverse systems have two things in common: (i) they are dissipative systems driven by homogeneous forcing, and (ii) there is range of forcing within which the application of different finiteamplitude perturbations can lead to distinct localized states. This proposal seeks to extend existing theory, partly developed by the PI and coworkers, to higher dimensions and to provide a comprehensive understanding of the effects of finite domain size, anisotropy and loss of translation invariance on the origin and properties of these structures. The techniques used include bifurcation theory for reversible and near-reversible systems, coupled with numerical branch-following and direct numerical simulations of realistic systems.Many systems respond to spatially uniform forcing by producing a spatial pattern. These patterns may take the form of a periodic array of cells or spots, or in particular cases by producing a single spot or group of spots. This proposal seeks to understand the relation between these two types of response, and to predict the conditions under which the latter response may be expected. There are many potential applications of a spot-like response. In optics individual spots may be "written" and "erased" using a laser beam, a procedure that may be used to store information. Mechanical structures often buckle in a localized way. Thin liquid films may break up into drops. A chemical process that uses catalysis may be degraded because the chemical conversion fails to proceed uniformly in space. These are all examples of spatially localized structures of the type that will be studied as part of this project. These spot-like structures may also move and interact. Such moving structures are involved, for example, in signaling along nerve fibers. The project seeks to predict the formation of these structures and to understand their basic properties.