Spatially Localized Structures in Higher Dimension

这是一个研究驱动耗散系统中空间局域结构的建议。这种结构的特征在于通过适当的强制和结构内的能量耗散的能量输入之间的平衡，并且在包括流体、非线性光学和反应扩散方程的许多物理系统中是常见的。该建议旨在通过对空间可逆系统中全局分叉的数学分析以及数值延拓技术，将现有对这些结构的理解扩展到二维和三维空间。将研究变分和非变分结构的系统，重点是了解参数变化时静止或运动结构的增长和多重性。 时间相关的局域化结构将通过分析体态和限制它的前沿或边界的运动来研究。运动结构之间的碰撞和它们被外部不均匀性捕获将通过直接数值模拟来研究。本文的结果将应用于几个表现出热驱动运动的系统，包括二元流体对流、双扩散对流和旋转对流。流体中常见的例子包括局部对流、涡旋、液滴和孤立波。另外的例子是光学和化学系统中的斑点，压缩下细长结构的局部屈曲，沿沿着神经纤维传播的脉冲和振动颗粒介质中的局部振荡。虽然这些例子涉及物理科学的许多领域，但出现的局部结构有许多共同的性质。该提案旨在发展对这种结构的起源和性质的数学理解，重点是随着系统参数的变化而发生的变化，并旨在确定这些不同例子所共有的性质。

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.