An algebro-analytic study on the trace formulas associated with the linear ordinary differential operators and the nonlinear integrable systems

阐明了与Dirac型算子相关的半交换算子的结构。特别是，三浦变换，这是孤立子理论的出发点，推广到整个平稳KdV族和相当有趣的身份被发现。另一方面，SIR模型，这是一个完全不同类型的动力系统，进行了研究。此外，利用迹公式，给出了构造与定常KdV族有关的第一积分全集的方案。利用第一积分，将与线性化算子相关的特征值问题的方程转化为Riemann球面上具有正则奇点的常微分方程，并利用该方程探讨了研究定常KdV族解的解析性质的方法.

The structure of the semi-commutative operators associated with the Dirac type operator is clarified. In particular, the Miura transformation, which is the starting point of the soliton theory, is generalized to the whole stationary KdV hierarchy and quite interesting identities are discovered. On the other hands, the SIR model, which is a dynamical system of quite different type, is studied. In addition, applying the trace formulas, the scheme for the construction of whole set of the first integrals associated with the stationary KdV hierarchy is obtained. Using the first integrals, the equation of the eigenvalue problem associated with the linearizing operator is transformed to the ordinary differential equation with the regular singular points on the Riemann sphere, and using that equation, the method to study the analytic properties of the solutions of the stationary KdV hierarchy is explored.