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

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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.