本项目研究一类与高阶矩阵谱问题相联系的多分量孤子方程Darboux变换的构造方法、并探讨Baker-Akhiezer函数及代数曲线在Darboux变换中的应用，期望得到拟周期背景下的精确解的一般构造方法。主要研究内容包括：（1）寻找和构造与高阶矩阵谱问题相联系的矩阵非线性演化方程，为可积系统提供更多的研究对象；（2）导出相应的Riccati方程，由此给出构造N重Darboux变换的基本方法，为可积系统的研究提供新的途径；（3）利用Riccati方程导出广义Darboux变换不依赖于取极限的显式表达式，显著减少广义Darboux变换的运算量；（4）探索利用Baker-Akhiezer函数处理拟周期种子解的一般方法，让代数曲线成为研究Darboux变换的新工具；（5）作为Darboux变换的应用，求得多分量可积方程的孤子、怪波、呼吸子、怪周期波等显式解，研究描述非线性现象的相互作用及动力行为。

In this project, we study the general approach to the construction of Darboux transformations for some multi-component soliton equations associated with high-order matrix spectral problems, and investigate the applications of Baker-Akhiezer functions and algebraic curves in Darboux transformations, expecting to find a method that produces new solutions under quasi-periodic backgrounds. In this program, we consider from the following five aspects. First, we find new integrable equations, enriching research subjects. Second, we derive Riccati equations from the Lax pairs, and then construct N-fold Darboux transformations on the basis of Riccati equations. Third, by revealing the relation between Riccati equations and general Darboux transformations, we write out exact formulas for high-order generalized Darboux transformations without involving taking limits, and then reduce the calculation complicity a lot. Fourth, we study a general method to study quasi-periodic seed solutions by using Baker-Akiezer functions and algebraic curves, providing a new tool for Darboux transformations. Fifth, we obtain explicit solitons solutions, rogue-wave solutions, breather solutions, rogue periodic solutions, etc., and study the interactions and dynamics of these nonlinear phenomena, aimed at a better understanding of nonlinear phenomena.