Painlev\'e type reductions for the non-Abelian Volterra lattices

TL;DR

This paper explores non-Abelian versions of the Volterra lattice, deriving Painlevé-type equations from symmetries, and presents new generalizations of discrete and continuous Painlevé equations.

Contribution

It introduces non-Abelian analogs of the Volterra lattice and derives Painlevé-type equations from their symmetries, expanding the understanding of integrable systems.

Findings

Derived non-Abelian Painlevé equations dP1 and dP34.

Obtained non-Abelian generalizations of P3, P4, and P5.

Connected lattice symmetries with Painlevé equations.

Abstract

The Volterra lattice admits two non-Abelian analogs that preserve the integrability property. For each of them, the stationary equation for non-autonomous symmetries defines a constraint that is consistent with the lattice and leads to Painlev\'e-type equations. In the case of symmetries of low order, including the scaling and master-symmetry, this constraint can be reduced to second order equations. This gives rise to two non-Abelian generalizations for the discrete Painlev\'e equations dP$_1$ and dP$_{34}$ and for the continuous Painlev\'e equations P$_3$, P$_4$ and P$_5$.