On a Toda lattice hierarchy: Lax pair, integrable symplectic map and algebraic-geometric solution

摘要

A Toda lattice hierarchy is studied by introducing a new spectral problem which is a discrete counterpart of the generalized Kaup-Newell spectral problem. Based on the Lenard recursion equation, Lax pair of the hierarchy is given. Further, the discrete spectral problem is nonlinearized into an integrable symplectic map. As a result, an algebraic-geometric solution in Riemann theta function of the hierarchy is obtained. Besides, two equations, the Volterra lattice and a (2+ 1)-dimensional Burgers equation with a discrete variable, yielded from the hierarchy are also solved.