Mappings preserving locations of movable poles:II. The third and fifth Painlevé equations

Abstract

In a recent paper(Gordoa P R et al 1999 Nonlinearity 12 955-68)we presented a new method of deriving Bäcklundtransformations (BTs) for ordinary differentialequations. The method is based on a consideration ofmappings that preserve a natural subset of movable poles,together with a careful asymptotic analysis of thetransformed equation, near each type of pole. In ouroriginal paper we applied this approach to the second andfourth Painlevé equations, and in a recent short paper(Gordoa P R et al 2001Glasgow Math. J. to appear) we gave preliminary results for the thirdand fifth Painlevé equations. Here we give fullresults for the latter two equations. For the thirdPainlevé equation withγδ¬ = 0 we obtain all fundamental BTs. We also obtain anew general formulation of the second iterates of BTs that includes bothγδ = 0 and γδ¬ = 0. For the fifth Painlevéequation we obtain all known (non-trivial) BTs. In addition, we obtain BTsrelating these two Painlevé equations, as well as BTs to equations ofsecond order and second degree, and special integrals.