Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P34 and P3

Abstract

We study reductions of the Korteweg–de Vries equation corresponding to stationary equations for symmetries from the noncommutative subalgebra. An equivalent system of n second-order equations is obtained, which reduces to the Painlevé equation P34 for n = 1. On the singular line t = 0, a subclass of special solutions is described by a system of n − 1 second-order equations, equivalent to the P3 equation for n = 2. For these systems, we obtain the isomonodromic Lax pairs and Bäcklund transformations which form the group Z2n×Zn.