The geometry of the classical solutions of the Garnier systems

Details The geometry of the classical solutions of the Garnier systems The geometry of the classical solutions of the Garnier systems

2001-06-25

arxiv.org/abs/math/0106208v2

Mathematics

Classical Analysis and ODEs

32 pages, Latex, 1 reference added

Scientific paper

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in $n$-variables, ${\cal G}_n$, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system ${\cal G}_n$. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system ${\cal G}_{n}$ is such that the associated Fuchsian system has $l$ monodromy matrices equal to $\pm\ID$, then it can be reduced classically to a solution of a the Garnier system with $n-l$ variables ${\cal G}_{n-l}$. When $n$ monodromy matrices are equal to $\pm\ID$, we have classical solutions of ${\cal G}_n$. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the Painlev\'e VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to $\pm\ID$.

Affiliated with

Also associated with

No associations

Rating

The geometry of the classical solutions of the Garnier systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The geometry of the classical solutions of the Garnier systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The geometry of the classical solutions of the Garnier systems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-194238