近年来，孤子理论和可积系统到非交换时空的延伸得到很多人的研究。非交换的一种自然解释是相空间的量子化致使通常的乘法为Moyal积取代。此外还有矩阵和四元数所引起的非交换性质。超对称方程由于费米量和玻色量的存在也可看作一类特殊的非交换可积系统。本项目所研究的非交换可积系统不考虑引起非交换的原因，只是形式地假设因变量AB≠BA，研究结果具有很强的普适性，适用于以上几种情形。拟行列式是针对除环上的矩阵定义的，它在非交换代数中的地位与行列式在交换代数中的地位同等重要。拟行列式在非交换可积系统、非交换对称函数、量子代数和Yangian等多个领域中都有重要应用，在一定程度上是非交换可积系统中'τ'函数的首选表示。本项目拟在已有工作基础上，利用Darboux变换等工具研究非交换可积系统与超对称方程的构造与拟行列式解，推广Sato理论到非交换情形，探索非交换可积系统的'τ'函数是否有拟Schur函数表示。

Recently, extension of soliton theory and integrable systems to noncommutative space-time has been studied by many authors. One natural way in which the variables fail to commute is because of a quantization of the phase space resulting in the normal product being replaced by a Moyal product. Besides, matrix and quaternion versions are other reasons giving rise to noncommutativity. Due to the existence of fermionic field variables and bosonic field variable, supersymmetric equations can be treated as a special kind of noncommutative integrable systems. In this project, we will not consider concrete reasons for noncommutativity, but formally assume dependent variables in a noncommutative integrable system are noncommutative, that is, AB≠BA. In this sense, our research results are of high generality and applicable to the above-mentioned cases. Quasideterminants are defined for matrices over division rings. It plays the same important role in noncommutative algebra as determinants in commutative algebra. Quasideterminants have been effective in many areas including noncommutative integrable systems, noncommutative symmetric functions, quantum algebra and Yangians. To a certain extent, quasideterminants should be regarded as preferred variables of τ functions in noncommutative integrable systems. Based on previous work, in this project, we are going to investigate construction and integrable properties, especially quasideterminant solutions of noncommutative integrable systems and supersymmetric equations by using Darboux transformations and other tool, extend Sato theory to noncommutative integrable systems and explore whether τ functions in noncommutative integrable systems could be expressed in terms of quasi-Schur functions or not.