随着科学技术的不断发展，人们越来越认识到自然界中一些非线性现象的重要性。相对于线性现象，非线性现象的性质更为复杂。作为非线性科学中一个重要组成部分，孤子理论在上世纪六十年代真正形成并成为一个非常活跃和富有吸引力的研究领域，它在数学、生物、大气海洋学及物理学等多个领域都有非常重要的应用。目前关于孤立子的研究已初步形成较完整的理论体系。近年来，孤子理论和可积系统到非交换情形的推广引起了人们极大的研究兴趣，一系列可积性质被揭示。非交换可积系统是可积系统的自然推广，同时具有重要的物理背景。国内外专家在可积系统与多项式函数的联系方面做了很多有意义的工作，建立了部分可积系统与Schur函数、正交多项式和Bell多项式的联系。本项目拟利用Darboux变换和Moutard变换研究非交换可积系统的Pfaffian解、与拟正交多项式和拟Schur型函数的联系以及q形变可积系统的q-Bell多项式理论。

With the development of science and technology, people start to realize the importance of nonlinearity phenomenon existing in nature. Compared to linearity, nonlinearity exhibits more complicated properties. As an important part of nonlinear science, soliton theory has formed since 1960s and become an active research area attracting many researchers into it. Soliton theory has important applications in many areas such as mathematics, biology, atmospheric and oceanic sciences, and physics. Some mathematical methods have been developed to deal with different aspects of integrable systems. As a natural generalization of commutative integrable systems, noncommutative integrable systems have important physical background. Consequently, a series of integrable properties are revealed so far. Some significant work has been done on the connections between integrable systems and Schur functions, orthogonal polynomials and Bell polynomials by international and domestic experts. In this research proposal, we are going to make use of Darboux transformations and Moutard transformations to study quasi-Pfaffian solutions to noncommutative integrable systems, connections of quasi-orthogonal polynomials and quasi-Schur functions with noncommutative integrable systems, applications of q-Bell polynomials to q-deformed integrable systems.