Algebro - Geometric constructions of the discrete Ablowitz - Ladik flows and applications

Resorting to the finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the discrete Ablowitz - Ladik equations and the (2 +1)-dimensional Toda lattice are decomposed into solvable ordinary differential equations. The straightening out of the continuous flow and the discrete flow is exactly given through the Abel - Jacobi coordinates. As an application, explicit quasiperiodic solutions for the (2+1)-dimensional Toda lattice are obtained. © 2003 American Institute of Physics.