Quasi-periodic solutions to the hierarchy of four-component Toda lattices

Starting from a discrete 3×3 matrix spectral problem, the hierarchy of four-component Toda lattices is derived by using the stationary discrete zero-curvature equation. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a trigonal curve Km−2Km−2 of genus m−2m−2 and present the related Baker–Akhiezer function and meromorphic function on it. Asymptotic expansions for the Baker–Akhiezer function and meromorphic function are given near three infinite points on the trigonal curve, from which explicit quasi-periodic solutions for the hierarchy of four-component Toda lattices are obtained in terms of the Riemann theta function.