Algebro-geometric initial value problems for discrete matrix modified Korteweg–de Vries equations: Tetragonal curves and Riemann theta function solutions

This paper develops the theory of tetragonal curves and introduces a systematic method for obtaining Riemann theta function solutions to algebro-geometric initial value problems for discrete matrix modified Korteweg–de Vries equations. In particular, the Lax pair of the discrete matrix modified Korteweg–de Vries hierarchy is derived from a discrete 4 × 4 matrix spectral problem using the discrete zero-curvature equation and Lenard equations. Then, we introduce the corresponding tetragonal curve and its Riemann theta function through the characteristic polynomial of the discrete Lax matrix, and also discuss the construction of three kinds of Abelian differentials. Building on the theory of tetragonal curves, we investigate algebro-geometric properties of Baker-Akhiezer functions and fundamental meromorphic functions. Finally, Riemann theta function solutions for the entire discrete matrix modified Korteweg–de Vries hierarchy are derived via asymptotic analysis.