Riemann theta function solutions to the semi-discrete Boussinesq equations

Abstract

The hierarchy of the semi-discrete Boussinesq equations associated with a discrete 4 × 4 matrix spectral problem has been derived by means of the zero-curvature and the Lenard recursion equations. The tetragonal curve is introduced by resorting to the characteristic polynomial of the Lax matrix for the semi-discrete Boussinesq hierarchy, upon which the Baker-Akhiezer functions, meromorphic functions, Abel differentials, and Riemann theta functions are constructed. Finally, we derive the Riemann theta function solutions to the semi-discrete Boussinesq hierarchy.