Quasi-periodic solutions for modified Toda lattice equation

Based on a spectral problem and the Lenard operator pairs, we derive in this paper a modified Toda lattice hierarchy. The modified Toda lattice equation is first decomposed into systems of integrable ordinary differential equations. A hyper-elliptic Riemann surface and Abel-Jacobi coordinates are then introduced to linearize the associated flow, from which some quasi-periodic solutions of the modified Toda lattice can be explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.