Trigonal Toda lattice Equation

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 $y(y-s)=x^3$, ($s\ne 0$).

9 pages, 3 figures