Spatially Localized Structures in Higher Dimension

这是研究驱动耗散系统中空间局域结构的一个建议。这种结构的特点是通过适当的强迫在能量输入和结构内的能量耗散之间取得平衡，并且在许多物理系统中很常见，包括流体，非线性光学和反应扩散方程。本提案旨在通过对空间可逆系统中的全局分岔的数学分析以及数值延拓技术，将对这些结构的现有理解扩展到二维和三维空间。将研究具有变分和非变分结构的系统，重点是理解随着参数变化而变化的静止或运动结构的增长和多样性。时间相关的局部结构将通过分析体状态和运动的前沿或边界，限制它进行研究。将采用直接数值模拟的方法研究运动结构之间的碰撞及其被外部不均匀性捕获的问题。结果将应用于几种热驱动运动系统，包括二元流体对流、双扩散对流和旋转对流。空间定域结构在物理系统中很常见。流体中常见的例子包括局部对流、涡旋、液滴和孤波。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以及振动颗粒介质中的局部振荡也提供了其他例子。虽然这些例子涉及物理科学的许多领域，但局部结构似乎有许多共同的性质。该提案旨在发展对这些结构的起源和性质的数学理解，重点关注随着系统参数变化而发生的变化，并旨在确定这些不同示例共享的属性。

This is a proposal to study spatially localized structures in driven dissipative systems. Such structures are characterized by a balance between energy input through suitable forcing and energy dissipation within the structure, and are common in many physical systems including fluids, nonlinear optics and reaction-diffusion equations. This proposal seeks to extend existing understanding of these structures to two and three spatial dimensions through mathematical analysis of global bifurcations in spatially reversible systems together with numerical continuation techniques. Systems with both variational and nonvariational structure will be studied, with a focus on understanding the growth and multiplicity of stationary or moving structures as parameters are varied. Time-dependent localized structures will be studied through analysis of the bulk state and the motion of the front or boundary that confines it. Collisions between moving structures and their trapping by external inhomogeneities will be investigated by direct numerical simulation. The results will be applied to several systems exhibiting thermally driven motion including binary fluid convection, doubly diffusive convection and rotating convection.Spatially localized structures are common in physical systems. Familiar examples in fluids include localized convection, vortices, liquid drops and solitary waves. Additional examples are provided by spots in optical and chemical systems, localized buckling of slender structures under compression, pulses propagating along neural fibers and localized oscillations in vibrating granular media. Although these examples reach across many areas of the physical sciences the localized structures that appear have many properties in common. This proposal seeks to develop a mathematical understanding of the origin and properties of such structures focusing on changes that take place as the parameters of the system change, and aims to identify the properties that are shared by these different examples.