2006 Fiscal Year Final Research Report Summary
The Painleve functions with solvable monodromy
| Project/Area Number |
16540191
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | Osaka University |
|---|
Principal Investigator |
OHYAMA Yosuke Osaka University, Graduate School of Information Science and Technology, Associate professor, 大学院情報科学研究科, 助教授 (10221839)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KAWANAKA Noriaki Osaka University, Graduate School of Information Science and Technology, Professor, 大学院情報科学研究科, 教授 (10028219)
HIBI Takayuki Osaka University, Graduate School of Information Science and Technology, Professor, 大学院情報科学研究科, 教授 (80181113)
SAKANE Yusuke Osaka University, Graduate School of Information Science and Technology, Professor, 大学院情報科学研究科, 教授 (00089872)
DATE Etsuro Osaka University, Graduate School of Information Science and Technology, Professor, 大学院情報科学研究科, 教授 (00107062)
NAGATOMO Kiyokazu Osaka University, Graduate School of Information Science and Technology, Associate professor, 大学院情報科学研究科, 助教授 (90172543)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | The Painleve equations / Monodromy / Hypergeometric equations |
|---|
| Research Abstract |
We constructed many examples of the Painleve functions, whose linear monodromy can be exactly calculated.
One example is 'algebraic solutions'. R. Fuchs proposed a problem "When does the linearized equation of the Painleve function reduce to a pull-back of the Gauss hypergeometric equation?" We gave an answer to this problem for the Painleve equations from the first to the fifth.
We also studied a coalescent diagram of the Painleve equations from the viewpoint of isomOnodromic deformations. We extended the coalescent diagram of the Painleve equations to the cases when the linearized equation contains irregular singular points whose Poincare rank are half-integers. Our coalescent diagram contains degenerated Painleve third equations, which are called D7 type and D8 type. The generic Painleve third equation is called D6 type. D6,D7 and D8 type equations contain 2,1and 0 parameters. We also studied degenerated Painleve third equations.
The second example of the Painleve functions with solvable monodromy is analytic solutions around fixed singular points.
For the third, fifth and sixth equations, we have
1)The number of analytic solutions around fixed singular points equals to the numbers of parameters
2)When we reduce the linearized equation in two ways, one reduced equation becomes trivial.
3)The linear monodromy is reduced to (confluent) hypergeometric equations.
These are special cases of Jimbo's connection formula. Our solution contains Umemura's classical solutions. When the solutions are algebraic around fixed singular points, we can also determine monodromy data.
|
