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.

聚合和集群模型中的动态缩放行为。 许多复杂的系统都表现出聚集行为，这些行为被观察到以明确的方式扩展，而当前的理论很难解释这些行为。 该计划是研究凝固模型中的动态尺度限制，这些模型（a）从根本上与分支过程和谱系相关，（b）描述相分离、混合连续生长和聚类，（c）模拟星际烟雾，以及（d）与染色体修复机制相关。 一个关键目标是发展重正化群思想，以加强以前用于研究标度对称系统的方法。 继梅农最近关于标量守恒定律中随机激波聚类的完全可积性的显着发现之后，工作将尝试识别和理解重要的解决方案族，并将该理论扩展到处理更一般的弹道聚合模型。流体和晶格中孤立波的存在性和稳定性理论。 孤立波是非线性科学中最早和最重要的相干结构研究之一。 最近的工作证明了小振幅孤立水波的线性稳定性，并开辟了新的研究途径。 使用空间加权范数来解释稳定性，空间加权范数测量远离非线性波的无穷小扰动的散射。 拟议的工作针对使用加权规范有望取得进展的问题，特别是：i）周期性介质中孤立波前的存在，包括多维粒子晶格和具有周期性测深的局域水波； ii) 三维水波 Kadomtsev-Petviashvili 模型的多维孤子前沿稳定性。