2013 Fiscal Year Final Research Report
An algebro-analytic study on the trace formulas associated with the linear ordinary differential operators and the nonlinear integrable systems
| Project/Area Number |
23540255
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Doshisha University |
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Principal Investigator |
OHMIYA Mayumi 同志社大学, 生命医科学部, 教授 (50035698)
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| Project Period (FY) |
2011 – 2013
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| Keywords | ダルブー変換 / ディラック型作用素 / ミウラ変換 / 定常KdV 階層 / 定常mKdV階層 / 準可換微分作用素 / 漸化作用素 / 跡公式 |
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| Research Abstract |
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.
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