Algebro-geometric initial value problems for integrable nonlinear lattices: Tetragonal curves and Riemann theta function solutions

In this paper, we establish the theory of tetragonal curves and address a series of fundamental problems within this framework, including the construction of a basis for holomorphic Abelian differentials, Abelian differentials of the second and third kinds, Baker-Akhiezer functions, and meromorphic functions. Building on these results, we apply the theory of tetragonal curves to investigate algebro-geometric initial value problems for integrable nonlinear lattice systems. As an illustrative example, we employ the discrete zero-curvature equation and the discrete Lenard equation to derive a hierarchy of coupled Bogoyavlensky lattice equations associated with a discrete $4\times 4$ matrix spectral problem. By analyzing the characteristic polynomial of the Lax matrix for this hierarchy, we introduce a tetragonal curve and its associated Riemann theta function, exploring the algebro-geometric properties of Baker-Akhiezer functions and a class of meromorphic functions. Using the Abel map and Abelian differentials, we precisely straighten out various flows. Finally, we obtain Riemann theta function solutions for the algebro-geometric initial value problems of the entire coupled Bogoyavlensky lattice hierarchy.