A systematic method for constructing time discretizations of integrable lattice systems: local equations of motion

TL;DR

This paper introduces a systematic, local method for discretizing time in integrable lattice systems that preserves their key properties and structures, demonstrated through various classical models.

Contribution

The authors develop a new local time discretization method based on zero-curvature representation, improving upon previous approaches by expressing auxiliary variables locally.

Findings

Preserves conserved quantities and solution structures of continuous systems.

Applicable to multiple integrable lattice models including Toda and Volterra.

Provides ultradiscrete analogues for certain lattice systems.

Abstract

We propose a new method for discretizing the time variable in integrable lattice systems while maintaining the locality of the equations of motion. The method is based on the zero-curvature (Lax pair) representation and the lowest-order "conservation laws". In contrast to the pioneering work of Ablowitz and Ladik, our method allows the auxiliary dependent variables appearing in the stage of time discretization to be expressed locally in terms of the original dependent variables. The time-discretized lattice systems have the same set of conserved quantities and the same structures of the solutions as the continuous-time lattice systems; only the time evolution of the parameters in the solutions that correspond to the angle variables is discretized. The effectiveness of our method is illustrated using examples such as the Toda lattice, the Volterra lattice, the modified Volterra lattice, the Ablowitz-Ladik lattice (an integrable semi-discrete nonlinear Schroedinger system), and the lattice Heisenberg ferromagnet model. For the Volterra lattice and modified Volterra lattice, we also present their ultradiscrete analogues.