# Some exact solutions of the Volterra lattice

## TL;DR

This paper investigates exact solutions of the Volterra lattice that satisfy a stationary condition, revealing their connection to Painlevé equations and expressing solutions via hypergeometric functions, with implications for initial data and solution regularity.

## Contribution

It establishes a link between stationary solutions of the Volterra lattice and Painlevé equations, providing explicit solutions and conditions for regularity.

## Key findings

- Solutions governed by Painlevé equations in time and space
- Explicit solutions expressed through confluent hypergeometric functions
- Characterization of initial data leading to regular solutions

## Abstract

We study solutions of the Volterra lattice satisfying the stationary equation for its non-autonomous symmetry. It is shown that the dynamics in $t$ and $n$ are governed by the continuous and discrete Painlev\'e equations, respectively. The class of initial data leading to regular solutions is described. For the lattice on the half-line, these solutions are expressed in terms of the confluent hypergeometric function. The Hankel transform of the coefficients of the corresponding Taylor series is computed on the basis of the Wronskian representation of the solution.