Title: On 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 Schrödinger (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 Schrödinger equation (DNLSE), which we
then use to produce integrable chain dynamics.