Algebro-geometric quasi-periodic solutions to the hierarchy of modified Bogoyavlensky lattice

Abstract

Under investigation is the modified Bogoyavlensky lattice, which serves as a generalization of the modified Volterra lattice. By introducing a discrete 3×3 matrix spectral problem, we propose an integrable hierarchy of the modified Bogoyavlensky lattice with the help of the zero-curvature equation and Lenard recursion relations. Based on the characteristic polynomial of Lax matrix for the hierarchy, we define a trigonal curve Km−1 of arithmetic genus m−1 and present the corresponding Baker–Akhiezer function and meromorphic function on it. By virtue of the asymptotic properties of the Baker–Akhiezer function and meromorphic function, we derive their explicit Riemann theta function representations. Algebro-geometric quasi-periodic solutions of the entire modified Bogoyavlensky hierarchy are obtained in terms of the asymptotic expansion of the meromorphic function and its Riemann theta function representation.