Volterra-Bogoyavlensky lattices and solutions of A2n(1) invariant Painlevé equations
Abstract
The objective of this work is to develop a framework that exploits the lattice structure of the k-th Volterra--Bogoyavlensky equations (k∈ℕ, k>1) to generate rational solutions of higher symmetric Painlevé equations. For k=2, we show that the Volterra lattice, equipped with suitable initial conditions, exactly models the one- and two-dimensional orbits generated by half-translation operators of the A2(1) symmetric Painlevé IV equations. This correspondence yields explicit closed-form expressions for all solution components in terms of generalized Okamoto polynomials and leads to new algebraic recurrence relations among these polynomials. We present two generalizations of the above Volterra lattice. One is derived from a fractional translation of the A4(1) symmetric Painlevé equations. It generalizes Volterra lattice structure in the multi-component setup of the affine A4(1) group and it is shown to generate solutions of the A4(1) symmetric Painlevé equations from the seed solution invariant under dihedral group D5. The other is the k=3 Bogoyavlensky lattice structure. It satisfies recurrence relations that naturally extend recurrence relations of the Volterra lattice. These results shed light on connection between Volterra--Bogoyavlensky lattices, dihedral symmetries, and rational solutions of higher Painlevé systems.
Versions
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BibTeX
@misc{adans2026volterrabogoyavlensky,
title = {{Volterra-Bogoyavlensky lattices and solutions of $A_2n^(1)$ invariant Painlevé equations}},
author = {Y. F. Adans and H. Aratyn and J. F. Gomes and G. V. Lobo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.13413},
eprint = {2606.13413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.13413},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Volterra-Bogoyavlensky lattices and solutions of A_2n^(1) invariant Painlevé equations
AU - Y. F. Adans
AU - H. Aratyn
AU - J. F. Gomes
AU - G. V. Lobo
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2606.13413
UR - https://arxiv.org/abs/2606.13413
AB - The objective of this work is to develop a framework that exploits the lattice structure of the k-th Volterra--Bogoyavlensky equations (k∈ℕ, k>1) to generate rational solutions of higher symmetric Painlevé equations. For k=2, we show that the Volterra lattice, equipped with suitable initial conditions, exactly models the one- and two-dimensional orbits generated by half-translation operators of the A_2^(1) symmetric Painlevé IV equations. This correspondence yields explicit closed-form expressions for all solution components in terms of generalized Okamoto polynomials and leads to new algebraic recurrence relations among these polynomials. We present two generalizations of the above Volterra lattice. One is derived from a fractional translation of the A_4^(1) symmetric Painlevé equations. It generalizes Volterra lattice structure in the multi-component setup of the affine A_4^(1) group and it is shown to generate solutions of the A_4^(1) symmetric Painlevé equations from the seed solution invariant under dihedral group D_5. The other is the k=3 Bogoyavlensky lattice structure. It satisfies recurrence relations that naturally extend recurrence relations of the Volterra lattice. These results shed light on connection between Volterra--Bogoyavlensky lattices, dihedral symmetries, and rational solutions of higher Painlevé systems.
ER -