Dynamics and stability in models of clustering and waves

聚集和聚集模型中的动态缩放行为。许多复杂的系统表现出聚集的行为，这些行为被观察到以定义明确的方式进行规模化，而目前的理论对这些行为的解释相当有限。该计划将研究凝血模型中的动态比例限制，这些模型(A)基本上与分支过程和谱系有关，(B)描述相分离、混合连续生长和聚集，(C)模拟星际烟雾，以及(D)与染色体的修复机制有关。一个关键的目标是发展重整化群思想，以加强以前用于研究标度对称系统的方法。在梅农最近发现了标量守恒律中随机激波簇集的完全可积性之后，工作将试图识别和理解重要的解族，并将该理论扩展到处理更一般的弹道聚集模型。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研究最早、最突出的相干结构之一。最近的工作证明了小振幅孤立水波的线性稳定性，并为研究开辟了新的途径。稳定性是用空间加权范数来解释的，该范数测量了远离非线性波的无限小扰动的散射。所提议的工作是针对使用加权范数有望取得进展的问题，特别是：i)周期介质中孤立波锋的存在，包括多维粒子格子和具有周期水深的局域水波；ii)三维水波的Kadomtsev-Petviashvili模型的多维孤子锋的稳定性。

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.