Volterra-Bogoyavlensky lattices and solutions of $A_{2n}^{(1)}$ invariant Painlev\'e equations

Abstract

The objective of this work is to develop a framework that exploits the lattice structure of the $k$-th Volterra--Bogoyavlensky equations ($k\in\mathbb N$, $k>1$) to generate rational solutions of higher symmetric Painlev\'e 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 Painlev\'e 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 Painlev\'e equations. It generalizes Volterra lattice structure in the multi-compneent setup of the affine $A_{4}^{(1)}$ group and it is shown to generate solutions of the $A_{4}^{(1)}$ symmetric Painlev\'e equations from the seed solutions 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.