2006 Fiscal Year Final Research Report Summary
Study of singularities of partial differential equations in the complex domain
| Project/Area Number |
16540169
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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 | Sophia University |
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Principal Investigator |
TAHARA Hidetoshi Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60101028)
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| Co-Investigator(Kenkyū-buntansha) |
OUCHI Sunao Sophia Univ., Fac. of Science and Technology, Professor, 理工学部, 教授 (00087082)
UCHIYAMA Koichi Sophia Univ., Fac. of Science and Technology, Professor, 理工学部, 教授 (20053689)
YOSHINO Kunio Sophia Univ., Fac. of Sci. and Tech., Associate Prof., 理工学部, 助教授 (60138378)
OKADA Yasunori Chiba Univ., Fac. of Science, Associate Prof., 理学部, 助教授 (60224028)
YAMANE Hideshi Kwansei Gauin Univ., Fac. of Science and Technology, Associate Prof., 理工学部, 助教授 (80286145)
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| Project Period (FY) |
2004 – 2006
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| Keywords | complex domain / partial differential equation / holomorphic solution / singularity / formal solution / equation of transformation |
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| Research Abstract |
1.In the theory of ODEs, the method of analytic transformation is often used to show the equivalence of two nonlinear ODEs. In this research, this method is extended to the framework of PDEs, and a equivalence of two nonlinear PDEs of Kowalewskian type is proved. The result is applied to the problem of analytic continuation of the solution.
2.It is proved that the formal solution of some class of nonlinear PDEs is multi-summable. The PDE in this class is regarded as the one obtained by the perturbation of ODEs. The result is applied to the normal form theory of vector fields.
3.A uniqueness result of the solution is proved for a class of nonlinear totally characteristic PDEs. The result is applied to showing the nonexistence of singularities of the solution.
4.Singularities of solutions of general nonlinear PDEs are studied and solutions with logarithmic singularities are constructed.
5.The asymptotic behaviour of solutions near the singularity is investigated and it is proved that in some classes of PDEs the asymptotic expansion is given concretely by using Mellin transformation. Gevrey type estimate of the remainder term is also obtained.
6.Singularities of sphere-symmetric solutions of p-elliptic equation are studied and the form of singularities is given explicitly by the analysis of Briot-Bouquet type PDEs.
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