| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2005: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2004: ¥1,500,000 (Direct Cost: ¥1,500,000)
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| Research Abstract |
To establish exact WKB analysis for Painleve hierarchies, we studied
1. the Stokes geometry of a higher order Painleve equation and its underlying Lax pair,
2. construction of formal solutions with free parameters to a higher order Painleve equation,
3. the structure of solutions near (simple) turning points,
and consequently obtained the following results.
Firstly, as for 1., we obtained a complete description of the Stokes geometry for the first Painleve hierarchy (whose Lax pair has the simplest structure). On the other hand, for Noumi-Yamada systems (whose Lax pair is of size greater than two) virtual turning points of the Lax pair are also relevant to the determination of the Stokes geometry of nonlinear equations, as is pointed out by S.Sasaki. The joint work with N.Honda is now clarifying that the roles of virtual turning points and new Stokes curves of the Lax pair can be well understood by introducing graph-theoretical notions such as "tree structure".
Secondly, as for 2., we succeeded in constructing instanton-type solutions by extending the method employed in the second order case, i.e., that of using reduction of a Hamiltonian system to its Birkhoff normal form, to higher order equations. Through this method we obtained formal solutions with free parameters for higher order Painleve equations that are expressible in the form of Hamiltonian systems like the first Painleve hierarchy.
Finally, as for 3., the structure theorem at a simple turning point of the first kind for 0-parameter solutions of Painleve hierarchies whose Lax pair is of size two was generalized to Noumi-Yamada systems.
Generalization of the structure theorem to instanton-type solutions and analysis at turning points of the second kind are important future problems; if they are overcome, connection problems for higher order Painleve equations will be solved explicitly.
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