Application of the trigonal curve to a hierarchy of generalized Toda lattices

Starting from the zero-curvature equation and Lenard recurrence relations, we derive a hierarchy of generalized Toda lattices. The trigonal curve is introduced through the Lax pair characteristic polynomial for the discrete hierarchy, from which a Dubrovin-type equation is established. Then the asymptotic behavior of the Baker–Akhiezer function and the meromorphic function is analyzed, and the divisors of the two functions are also discussed. Moreover, the Abel map is defined and the corresponding flows are straightened out on the Jacobian variety, such that the final algebro-geometric solutions of the hierarchy are calculated in terms of the Riemann theta function.