Mathematical Sciences: Mathematical and Statistical Foundations of Chaotic Solitons

点击翻译按钮获取中文摘要

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.