On Energy Functions for String-Like Continuous Curves, Discrete Chains, and Space-Filling One Dimensional Structures

TL;DR

This paper develops a geometric framework to derive energy functions for string-like curves and chains, ensuring invariance under local frame rotations and reparametrizations, and explores integrability and dual hierarchies in both continuous and discrete settings.

Contribution

It introduces a systematic method to construct energy functions based on invariance principles, including a dual hierarchy for the nonlinear Schrödinger equation, and extends these concepts to discrete chains.

Findings

Energy functions are derived from conserved quantities of the NLSE hierarchy.

A dual hierarchy via Weyl transformation is proposed and shown to be integrable.

Discrete energy functions converge to continuum counterparts in the limit.

Abstract

The theory of string-like continuous curves and discrete chains have numerous important physical applications. Here we develop a general geometrical approach, to systematically derive Hamiltonian energy functions for these objects. In the case of continuous curves, we demand that the energy function must be invariant under local frame rotations, and it should also transform covariantly under reparametrizations of the curve. This leads us to consider energy functions that are constructed from the conserved quantities in the hierarchy of the integrable nonlinear Schr\"odinger equation (NLSE). We point out the existence of a Weyl transformation that we utilize to introduce a dual hierarchy to the standard NLSE hierarchy. We propose that the dual hierarchy is also integrable, and we confirm this to the first non-trivial order. In the discrete case the requirement of reparametrization invariance is void. But the demand of invariance under local frame rotations prevails, and we utilize it to introduce a discrete variant of the Zakharov-Shabat recursion relation. We use this relation to derive frame independent quantities that we propose are the essentially unique and as such natural candidates for constructing energy functions for piecewise linear polygonal chains. We also investigate the discrete version of the Weyl duality transformation. We confirm that in the continuum limit the discrete energy functions go over to their continuum counterparts, including the perfect derivative contributions.