Mathematical Sciences: Mathematical and Statistical Foundations of Chaotic Solitons

这项研究构成了建立新的数学工具和计算模型的跨学科和国际化的方法，用于研究具有孤子统计行为的定域和非平稳混沌吸引子。这些吸引子被称为混沌孤子，理论上认为它们在简单的保守系统中发挥作用，与一系列现象直接相关，包括基本粒子相互作用、原子聚集和生物分子动力学，如蛋白质折叠和细胞骨架组装。混沌和孤子通常被描绘成不同的，结果是量子理论和混沌通常是分离的。然而，目前的研究方向是量子场论和多维非稳态孤子之间的数学联系；这些联系表明描述量子过程的更简单的方法，可以从经典的混沌和网络理论派生出来。将探索用于表示吸引子场态的统计描述的代数和拓扑方法。将一维混沌孤子模型扩展到具有与观测物理相关的约束的二维、三维和四维孤子模型将是基本的第一步。俄罗斯杜布纳联合核研究所的一个长期数学家小组在I.博格鲁布斯基博士的指导下，完成了大量的理论工作。这项建议的一个主要目标是巩固、评估这个俄罗斯团队的工作，并以数学的方式将其转化为美国数学、统计和物理学领域易于使用的数学和计算机建模。拟议的努力是基于这样一种信念，即通过国际合作研究组织努力巩固和扩大这项工作是非常紧迫的。

This investigation constitutes an interdisciplinary and international approach to establishing new mathematical tools and computational models for the study of localizable and nonstationary chaotic attractors with the statistical behavior of solitons. These attractors, termed chaotic solitons, have been theorized as having a role in simple conservative systems with direct relevance to a wide range of phenomena including elementary particle interactions, atomic clustering, and biomolecular dynamics such as protein folding and cytoskeletal assembly. Chaos and solitons have generally been portrayed as being disparate with the consequence that quantum theory and chaos have been generally dissociated. However, the present line of research points toward mathematical affinities between quantum field theory and multidimensional nonstationary solitons; these affinities indicate simpler methods for describing quantum processes, derivable from classical chaos and network theories. Algebraic and topological methods for representing the statistical descriptions of attractor field states will be explored. Extension of one-dimensional chaotic soliton models into two, three and four dimensions with constraints relevant to observed physics will be a fundamental first step. An extensive amount of theoretical work has been done in particular by a long-standing group of mathematicians at the Joint Institute for Nuclear Research in Dubna, Russia, under the direction of Dr. I. Bogolubsky. A major goal of this proposal is to consolidate, evaluate, and mathematically transform the work that has grown out of this Russian team into mathematics and computer modeling readily accessible to the U. S. mathematical, statistical, and physics communities. The proposed effort is based on the belief that it is time-critical to organize efforts to consolidate and extend this work through international cooperative study.