Application of tetragonal curves theory to the 4-field Błaszak–Marciniak lattice hierarchy

Abstract

Through the paper, we explore the theory of tetragonal curves and derive the quasi-periodic solutions to the 4-field Błaszak–Marciniak lattice hierarchy. The hierarchy associated with a discrete fourth-order matrix spectral problem is derived from the zero-curvature equation and discrete Lenard equation. The tetragonal curve and its related Riemann theta functions are introduced through the characteristic polynomial of the Lax matrix. Additionally, the Baker-Akhiezer functions and a class of meromorphic functions on the tetragonal curve are investigated. Furthermore, the Abel map and Abelian differentials are used to straighten out various flows, leading ultimately to the quasi-periodic solutions of the 4-field Błaszak–Marciniak lattice hierarchy.