The Integrable Dynamics of Discrete and Continuous Curves

TL;DR

This paper demonstrates that certain geometric properties of curve motion on spheres lead to integrable dynamics, unifying continuous and discrete cases and deriving well-known integrable equations like nonlinear Schrödinger and sine-Gordon.

Contribution

It introduces a geometric framework that characterizes integrable dynamics of curves on spheres, encompassing both continuous and discrete cases, and derives classical integrable equations within this setting.

Findings

Derivation of nonlinear Schrödinger and sine-Gordon equations from geometric properties.

Identification of geometric conditions that ensure integrability of curve dynamics.

Unification of continuous and discrete integrable curve evolutions within a common framework.

Abstract

We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch during the motion, iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. Well known examples of integrable evolution equations, like the nonlinear Schroedinger and the sine-Gordon equations, as well as their discrete analogues, are derived in this general framework.