On the geometry of the first and second Painlevé equations

Details On the geometry of the first and second Painlevé equations On the geometry of the first and second Painlevé equations

2007-11-18

arxiv.org/abs/0711.2815v7

Mathematics

Differential Geometry

The research was supported in part by the Czech Ministry of Education, Youth and Sports within the project LC06002

Scientific paper

In this paper we \emph{explicitly} compute the transformation that maps the generic second order differential equation $y''= f(x, y, y')$ to the Painlev\'e first equation $y''=6y^2+x$ (resp. the Painlev\'e second equation ${y''=2 y^{3}+yx+ \alpha}$). This change of coordinates, which is function of $f$ and its partial derivatives, does not exist for every $f$; it is necessary that the function $f$ satisfies certain conditions that define the equivalence class of the considered Painlev\'e equation. In this work we won't consider these conditions and the existence issue is solved \emph{on line} as follows: If the input equation is known then it suffices to specialize the change of coordinates on this equation and test by simple substitution if the equivalence holds. The other innovation of this work lies in the exploitation of discrete symmetries for solving the equivalence problem.

Affiliated with

Also associated with

No associations

Rating

On the geometry of the first and second Painlevé equations 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 On the geometry of the first and second Painlevé equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the geometry of the first and second Painlevé equations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-297716