2002 Fiscal Year Final Research Report Summary
Study of Diophantine Phenomena Appearing in Asymptotic Analysis of Nonlinear Partial Differential Equations
| Project/Area Number |
11640183
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | CHUO UNIVERSITY |
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Principal Investigator |
YOSHINO Masafumi Faculty of Economics, Chuo University, Professor, 経済学部, 教授 (00145658)
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| Project Period (FY) |
1999 – 2002
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| Keywords | nonlinear Fuchsian system / Diophantine condition / resonance / Frobenius theorem / singular solution / asymptotic solution / simultaneous normal forms / WKB method |
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| Research Abstract |
We have obtained two main results 1) and 2).
1) We reveal the role of Diophantine conditions when solving a system of nonlinear partial differential equations or difference equations in the normal form theory. Namely we have
a) The simultaneous normal form of a commuting system of maps under Bruno condition is presented. This is also valid for a system of circle maps in a Gevrey class.
b) The solvability of nonlinear Fuchsian equations of general variables in a class of finitely smooth functions is proved. This yields a so-called Grobman-Hartmann type theorem.
c) The global normal form of a pseudodifferential operator on tori which is a perturbation of a system of constant vector fields is presented. Necessary and sufficient conditions are given in terms of a Diophantine condition.
d) To make clear the relations between a simultaneous Diophantine condition of a commuting system of maps or vector fields and the Diophantine conditions for every generator of the system.
2) We show that a Riemann-Hilbert problem is closely related to the solvability of equations without Diophantine conditions. Namely we have
a) Solvability of nonlinear Fuchsian equations of general variables under a so-called Riemann-Hilbert condition is proved. This is applied to a mixed Monge- Ampere equation.
b) Solvability of the same equations as in a) in a class of singular functions under a Poincare condition is presented. This can be regarded as a Frobenius theorem for a partial differential equation. The results are partly reported in ICM conference in Beijing in 2002.
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Research Products
(16 results)
