Dynamics and stability in models of clustering and waves

聚集和聚类模型中的动态标度行为。 许多复杂系统表现出聚集行为，这些行为被观察到以定义明确的方式扩展，而目前的理论解释得很差。 该计划是研究凝固模型的动态标度限制，（a）从根本上与分支过程和谱系有关，（B）描述相分离，混合连续增长和集群，（c）星际烟雾模型，（d）与染色体中的修复机制有关。 一个关键的目标是发展重整化群的思想，加强以前用于研究标度对称系统的方法。 继显着最近发现梅农有关完全可积性随机冲击集群标量守恒定律，工作将试图确定和理解重要的家庭的解决方案，并扩展理论，以处理更一般的模型弹道aggregation.Existence和稳定性理论孤立波在流体和晶格。 孤立波是非线性科学中研究最早、最突出的一类相干结构。 最近的工作提供了一个小振幅孤立波的线性稳定性的证明，并开辟了新的研究途径。 稳定性的解释，使用空间加权范数，测量散射的无穷小扰动远离非线性波。 建议的工作是针对问题的加权规范的使用承诺产生的进展，特别是：i）存在孤立波阵面在周期性介质中，包括多维粒子晶格和局部水波与周期水深;和ii）多维孤立波阵面的稳定性的Kadomtsev-Petviashvili模型的三维水波。

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.