Algebro–geometric constructions of the discrete Ablowitz–Ladik flows and applications

Abstract

Resorting to the finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the discrete Ablowitz–Ladik equations and the $\text{(2+1)}$ -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 $\text{(2+1)}$ -dimensional Toda lattice are obtained.