# Trigonal Toda lattice Equation

## TL;DR

This paper introduces a new form of the Toda lattice equation on a trigonal lattice and explores its elliptic solutions related to a specific elliptic curve, expanding the understanding of integrable systems on complex lattices.

## Contribution

It formulates the trigonal Toda lattice equation on a complex lattice and derives its elliptic solutions associated with a particular elliptic curve.

## Key findings

- Derived the trigonal Toda lattice equation on a 6-regular graph.
- Established elliptic solutions related to the curve y(y-s)=x^3.
- Extended Toda lattice theory to complex trigonal lattice structures.

## Abstract

In this article, we give the trigonal Toda lattice equation, $$ -\frac{1}{2}\frac{d^3}{d t^3} q_{\ell}(t) = e^{q_{\ell+1}(t)} +e^{q_{\ell+\zeta_3}(t)} +e^{q_{\ell+\zeta_3^2}(t)}-3e^{q_\ell(t)}, $$ for a lattice point $\ell \in \mathbb{Z}[\zeta_3]$ as a directed 6-regular graph where $\zeta_3=e^{2\pi i/3}$, and its elliptic solution for the curve $y(y-s)=x^3$, ($s\neq 0$).