本项目研究多分量可积系统的Liouville相关性，并在此基础上，研究其可积性质、几何结构和色散量子化。首先，通过联系谱问题的Liouville变换，建立多分量可积方程族的Liouville相关性；发展和完善Liouville相关性的理论框架；研究由此得到的新的多分量可积系统的可积结构。其次，研究联系两类具有不同色散结构的可积方程族的Liouville变换的几何意义，以及相应可积方程族的几何结构及其可积性质的几何特征。第三，研究多分量可积系统行波解的分类，孤立子解的存在性和形成机理；分析具有不同类型色散结构的可积系统的行波解在Liouville变换下的联系。最后，研究具有线性色散结构的可积系统的色散量子化现象；探讨色散量子化效应在Liouville变换下的行为，挖掘允许非线性色散结构可积系统的与Talbot效应对应的，不同于行波解的性态。

This project concerns with the Liouville correspondence, and on the basis of which, the integrable properties, the geometric formulations and dispersive quantization for the multi-component integrable systems. Firstly, we establish the Liouville correspondence between the multi-component integrable hierarchies by utilizing the Liouville transformation between their respective isospectral problems, systematize the theory of Liouville correspondence for multi-component integrable systems and explore the integrable structures for the newly discovered integrable systems. Next, we study the geometric significance of the Liouville transformation between two integrable hierarchies endowed with disparate dispersive structures, and further explore the geometric structures of integrable systems possessing Liouville correspondence, with the aim to analyze the geometric formulations of their integrability. Thirdly, we classify the analytic and non-analytic traveling wave solutions, the existence and the formation mechanism of solition solutions to the multi-component integrable systems, and further analyze the relationship between the traveling wave solutions admitted by the multi-component integrable systems exhibiting different types of dispersive characteristic. Finally, we investigate the Talbot effect of dispersive quantization and fractalization for the nonlinear integrable systems with linear dispersion, and provide formulation of theorems and rigorous proofs. In particular, we aim to identify the effect of the Liouville transformation on the qualitative behavior of the solutions with the dispersive quantization characteristic, hoping to unfold the qualtitative features of those solutions which correspond to the nonlinear dispersive Talbot effect and differ with the traveling wave solutions, and to find undiscovered genuinely phenomena for the nonlinear dispersive integrable systems.