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

V. E. Adler; M. P. Kolesnikov

doi:10.1063/5.0091939

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arXiv · 2203.09977

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

V. E. Adler·M. P. Kolesnikov

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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.

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V. E. Adler, M. P. Kolesnikov. 2022-03-18. Non-Abelian Toda lattice and analogs of Painlev\'e III equation. https://doi.org/10.1063/5.0091939

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