Relation between discrete Frenet frames and the bi-Hamiltonian structure of the discrete nonlinear Schrödinger equation

Abstract

The discrete Frenet equation entails a local framing of a discrete, piecewise linear polygonal chain in terms of its bond and torsion angles. In particular, the tangent vector of a segment is akin the classical O(3) spin variable. Thus there is a relation to the lattice Heisenberg model, that can be used to model physical properties of the chain. On the other hand, the Heisenberg model is closely related to the discrete nonlinear Schr\"odinger (DNLS) equation. Here we apply these interrelations to develop a perspective on discrete chains dynamics: We employ the properties of a discrete chain in terms of a spinorial representation of the discrete Frenet equation, to introduce a bi-hamiltonian structure for the discrete nonlinear Schr\"odinger equation (DNLSE), which we then use to produce integrable chain dynamics.