Non-Abelian Toda lattice and analogs of Painlev\'e III equation

TL;DR

This paper explores non-Abelian generalizations of the Toda lattice and Painlevé III equations, revealing new integrable structures and reductions through stationary equations and self-similar transformations.

Contribution

It introduces non-Abelian analogs of classical integrable systems and demonstrates their derivation via stationary and self-similar reductions.

Findings

Non-Abelian Toda lattice reduces to a non-autonomous system similar to Maxwell–Bloch equations.

Self-similar reduction yields non-Abelian Painlevé III equations.

The Toda lattice acts as an auto-Bäcklund transformation for these systems.

Abstract

In integrable models, stationary equations for higher symmetries serve as one of the main sources of reductions consistent with dynamics. We apply this method to the non-Abelian two-dimensional Toda lattice. It is shown that already the stationary equation of the simplest higher flow gives a non-trivial non-autonomous constraint that reduces the Toda lattice to a non-Abelian analog of the pumped Maxwell--Bloch equations. The Toda lattice itself is interpreted as an auto-B\"acklund transformation acting on the solutions of this system. Further self-similar reduction leads to non-Abelian analogs of the Painlev\'e III equation.