Discrete Local Induction Equation

TL;DR

This paper introduces a discrete analogue of the local induction equation, modeling the deformation of discrete space curves via the discrete nonlinear Schrödinger equation, with explicit formulas using tau functions of the KP hierarchy.

Contribution

It develops a novel discrete model for space curve deformation based on the discrete nonlinear Schrödinger equation, extending the continuous theory.

Findings

Formulation of a discrete local induction equation.

Explicit tau function formulas for discrete curves.

Connection to the two-component KP hierarchy.

Abstract

The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schr\"odinger equation. In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schr\"odinger equation. We also present explicit formulas for both smooth and discrete curves in terms of $\tau$ functions of the two-component KP hierarchy.