Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P${}_{34}$ and P${}_3$

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 P${}_{34}$ 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 P${}_3$ equation for $n=2$. For these systems, we obtain the isomonodromic Lax pairs and Bäcklund transformations which form the group ${\mathbb{Z}}_2^n\times {\mathbb{Z}}^n$.

arXiv comment11 pages

References in corpus (4)

• On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
• Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach
• The Hamiltonian Structure of the Second Painleve Hierarchy
• Nonautonomous symmetries of the KdV equation and step-like solutions

Cited by in corpus (4)

• Negative flows for several integrable models
• 3D-consistency of negative flows
• Negative flows and non-autonomous reductions of the Volterra lattice
• Vector systems of Painlevé type