Dynamics and stability in models of clustering and waves

聚类和波模型中的动力学和稳定性

• 批准号: 1211161
• 负责人: Robert Pego
• 金额: $ 29.16万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211161&HistoricalAwards=false
• 关键词: Dynamics; stability; models; clustering; waves

Dynamic scaling behavior in models of aggregation and clustering. Numerous complex systems exhibit clumping behaviors that are observed to scale in well-defined ways that are rather poorly explained by current theory. The plan is to study dynamic scaling limits in coagulation models that (a) are fundamentally related to branching processes and genealogy, (b) describe phase separation, mixing continuous growth and clustering, (c) model interstellar smoke, and (d) relate to repair mechanisms in chromosomes. A key goal is to develop renormalization-group ideas that strengthen methods previously used to study scaling-symmetric systems. Following the remarkable recent discovery of Menon concerning complete integrability for random shock clustering in scalar conservation laws, work will try to identify and understand important families of solutions, and extend the theory to handle more general models of ballistic aggregation.Existence and stability theory for solitary waves in fluids and lattices. Solitary waves are one of the earliest and most prominent kinds of coherent structures to be studied in nonlinear science. Recent work has provided a proof of linear stability for solitary water waves of small amplitude, and has opened up new avenues for investigation. Stability is explained using spatially weighted norms that measure the scattering of infinitesimal perturbations away from the nonlinear wave. The proposed work is directed toward issues for which the use of weighted norms promises to produce progress, especially: i) existence of solitary wave fronts in periodic media, including multi-dimensional particle lattices and localized water waves with periodic bathymetry; and ii) stability of multidimensional soliton fronts for the Kadomtsev-Petviashvili model of three-dimensional water waves.

聚集和聚类模型中的动态缩放行为。 许多复杂的系统都表现出凝结行为，这些行为被观察到以明确的方式进行扩展，而当前理论对这些行为的解释很差。 该计划是在凝血模型中研究动态缩放限制，该模型（a）与分支过程和家谱基本上相关，（b）描述相位分离，混合持续生长和聚类，（c）模型星际烟雾，以及（d）与染色体中的修复机制有关。 一个关键目标是开发重新归一化组的思想，以增强先前用于研究缩放对称系统的方法。 在最近发现了Menon关于标量保护法中随机冲击聚类的完全集成性的显着发现之后，工作将尝试识别和理解解决方案的重要家庭，并扩展理论以处理弹道聚集的更一般的模型。存在液体和稳定性的流体和稳定性理论。 孤独的波浪是在非线性科学中研究的最早，最突出的连贯结构之一。 最近的工作为小幅度的孤立水波提供了线性稳定性的证明，并为研究开辟了新的途径。 使用空间加权规范来解释稳定性，该规范测量了无限扰动的散射，远离非线性波。 拟议的工作针对使用加权规范有望产生进展的问题，特别是：i）周期性介质中存在孤立波阵线，包括多维颗粒晶格和周期性测深的局部水波； ii）三维水波的Kadomtsev-Petviashvili模型的多维孤子前部的稳定性。