Algebro-geometric integration of the modified Belov–Chaltikian lattice hierarchy

Using the Lenard recurrence relations and the zero-curvature equation, we derive the modified Belov–Chaltikian lattice hierarchy associated with a discrete $3\times3$ matrix spectral problem. Using the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a trigonal curve $\mathcal{K}_{m-2}$ of arithmetic genus $m-2$. We study the asymptotic properties of the Baker–Akhiezer function and the algebraic function carrying the data of the divisor near $P_{\infty_1}$, $P_{\infty_2}$, $P_{\infty_3}$, and $P_0$ on $\mathcal{K}_{m-2}$. Based on the theory of trigonal curves, we obtain the explicit theta-function representations of the algebraic function, the Baker–Akhiezer function, and, in particular, solutions of the entire modified Belov–Chaltikian lattice hierarchy.