Localized Structures in Spatially Extended Systems: Fronts and Defects

空间局域结构如前锋、缺陷、斑点或脉冲在许多连续介质系统中是常见的，包括沿神经纤维传播的脉冲、光学和化学系统中的耗散孤子、细长结构在压缩下的局域屈曲以及振动颗粒介质中的振荡。流体的例子包括局部对流、漩涡和液滴。这些不同的系统有两个共同点：(I)它们是由空间均匀强迫驱动的耗散系统，(Ii)在不同的强迫范围内，应用不同的有限振幅扰动可以导致不同的局域态。研究人员试图将现有理论扩展到新的方向，重点关注材料科学中出现的问题，如过冷液体中晶体的成核和生长，以及可能导致的有序和无序结构。这包括被称为准晶的具有短程有序但没有长程有序的结构。所形成的结构的类型反过来决定了所得到的材料的强度和其他性能，这在很大程度上取决于结晶过程的速度。该项目的目的是全面了解在该系统和相关系统中液体在不同温度下发生的不同类型生长背后的机制。两名博士后参与了该项目的研究。在这个项目中，研究人员和他的同事们研究了二维和三维空间局域结构的性质。考虑了守恒和非守恒系统，重点讨论了软物质从过冷熔体结晶的相场和动力学密度泛函理论模型。根据这一过程的速度，所得到的材料可能是结晶的、非晶态的或准晶的。每种类型的局域结构都可以作为晶体成核的关键核，因此特别有意义。该项目的重点是这些结构的特性，以及将它们与熔体分开的前锋的特性。同时考虑了定常波阵面和传播波阵面，并详细研究了决定传播速度的过程。其中包括锋面与其后面的微观结构的钉扎以及与守恒质量模的相互作用。该项目还包括对非保守系统中的行波脉冲的平行研究，以及对不同类型的缺陷的详细研究，其中钉扎微结构是重要的。研究人员使用的技术包括分叉理论，结合数值分支跟踪和现实系统的直接数值模拟。两名博士后学生参与了该项目的研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Spatially localized structures such as fronts, defects, spots or pulses are common in many continuum systems, and include pulses propagating along nerve fibers, dissipative solitons in optical and chemical systems, localized buckling of slender structures under compression, and oscillons in vibrating granular media. Examples from fluids include localized convection, vortices and drops. These diverse systems have two things in common: (i) they are dissipative systems driven by spatially uniform forcing, and (ii) there is range of forcing within which the application of different finite amplitude perturbations can lead to distinct localized states. The investigator seeks to extend existing theory in new directions, focusing on problems arising in materials science such as the nucleation and growth of crystals from a supercooled liquid, and the ordered and disordered structures that may result. These include structures with short-range order but no long-range order called quasicrystals. The type of structure that forms in turn determines the strength and other properties of the resulting material and this depends strongly on the speed of the crystallization process. The aim of the project is to provide a comprehensive understanding of the mechanisms behind the different types of growth that take place at different temperatures of the liquid both in this and in related systems. Two postdoctoral students are engaged in the research of the project.In this project the investigator and his colleagues study the properties of spatially localized structures in two and three dimensions. Both conserved and nonconserved systems are considered with a focus on phase field and dynamical density functional theory models of soft matter crystallization from a supercooled melt. Depending on the speed of this process, the resulting material may be crystalline, amorphous, or a quasicrystal. Localized structures of each type may serve as critical nuclei for the nucleation of the crystal and are therefore of particular interest. The project focuses on the properties of these structures, and the properties of the fronts that separate them from the melt. Both steady and propagating fronts are considered and the processes that determine the propagation speed are studied in detail. These include pinning of fronts to the microstructure behind them and the interaction with the conserved mass mode. The project also includes a parallel study of traveling pulses in nonconserved systems and a detailed study of different types of defects where pinning to microstructure is important. The techniques used by the investigator include bifurcation theory, coupled with numerical branch-following and direct numerical simulations of realistic systems. Two postdoctoral students are engaged in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.