dNLS Flow on Discrete Space Curves

TL;DR

This paper introduces a discrete analogue of the binormal flow on space curves, linking discrete space curve deformation to the discrete nonlinear Schrödinger equation and providing explicit formulas via the KP hierarchy.

Contribution

It presents the discrete nonlinear Schrödinger flow on discrete space curves and derives explicit formulas using the tau function of the 2-component KP hierarchy.

Findings

Discrete space curve deformation modeled by dNLS.

Explicit formulas for flows using KP hierarchy tau functions.

Bridges continuous and discrete integrable curve dynamics.

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 (NLS). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schr\"odinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the $\tau$ function of the 2-component KP hierarchy.