Validity of Soliton-Picture in Nonlinear Phenomena

在孤子理论及其推广的基础上，本文进行了以下研究：研究了弯曲和扭转两种情况下的一维弹性。考虑到环状dna的拓扑约束，我们得到了超卷曲dna的宏观模型。将孤子理论扩展到量子系统，我们可以解释经典孤子是如何从量子孤子中产生的。该公式也可用于描述量子孤子在外力作用下的分裂。量子逆散射方法为研究动力学和统计力学中的精确可解模型提供了一个统一的框架。关键是Yang-Baxter关系，它是解的充分条件。通过求解Yang-baxter关系，我们发现二维经典统计力学中存在至少可解的模型。从统计力学中精确可解的模型中导出结点和连杆的拓扑不变量连杆多项式的一般方法。以琼斯多项式为最简形式，得到了一组新的、功能强大的连杆多项式。还发现了双变量扩展和新的代数。

Based on the soliton theory and its generalizations, the following researches have been done.1. One-dimensional elasticity is studied with both bending and torsion. Taking account of topological constraints for circular DNAs we obtain a macroscopic model for supercoiling DNAs.2. Extending the soliton theory to quantum systems we can expl-ain how classical solitons arise from quantum solitons. The formulation is also useful to describe break-up of quantum soliton under external forces.3. The quantum inverse scattering method yiels a unified frame-work to study exactly solvable models in dynamics and statis-tical mechanics. The key is the Yang-Baxter relation which is a sufficient condition for the solvability. By solving the Yang-baxter relation, we find that there exist at least solvable models in two-dimensional classical statis-tical mechanics.4. A general method is found to derive link polynomial, topolo-gical invariant for knots and links, from an exactly solvable model in statistical mechanics. A set of new and powerful link polynomials, which includes the Jones polynomial as the simplest case, is found. Two-variable extensions and new algebras are also found.