Mathematical Sciences: Mathematical and Statistical Foundations of Chaotic Solitons

本研究为研究具有孤子统计行为的可定域非稳态混沌吸引子建立了新的数学工具和计算模型。这些吸引子，被称为混沌孤子，已经被理论化为在简单的保守系统中发挥作用，与广泛的现象直接相关，包括基本粒子相互作用，原子聚类和生物分子动力学，如蛋白质折叠和细胞骨架组装。混沌和孤子通常被描述为完全不同的，结果是量子理论和混沌通常是分离的。然而，目前的研究方向指向量子场论与多维非平稳孤子之间的数学联系；这些相似性表明了描述量子过程的更简单的方法，可以从经典混沌和网络理论中推导出来。将探讨表示吸引子场态统计描述的代数和拓扑方法。将一维混沌孤子模型扩展到二维、三维和四维，并与观测到的物理相关的约束将是基本的第一步。俄罗斯杜布纳联合核研究所（Joint Institute for Nuclear Research）的一群数学家，在I. Bogolubsky博士的指导下，已经完成了大量的理论工作。该提案的一个主要目标是巩固、评估和在数学上将这个俄罗斯团队的成果转化为易于美国数学、统计和物理社区访问的数学和计算机建模。所提议的努力是基于这样一种信念，即通过国际合作研究来组织巩固和扩大这项工作的努力是非常及时的。

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.