Energy functions for stringlike continuous curves, discrete chains, and space-filling one dimensional structures

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 Schrödinger equation. We point out the existence of a Weyl transformation that we utilize to introduce a dual hierarchy to the standard nonlinear Schrödinger equation hierarchy. We propose that the dual hierarchy is also integrable, and we confirm this to the first nontrivial 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.

Figure 1

Schematic relations between those conserved densities that have a natural interpretation in terms of gauge field theory, in the positive-order (55) and negative-order (59, 61) hierarchies. AHM/NLSE is for the Abelian Higgs Model, a.k.a. the nonlinear Schrödinger equation, Weyl denotes a relation by the Weyl duality between the hierarchies, and Z-S stands for the Zakharov-Shabat recursive relation. Both hierarchies amalgamate at $H_0=0$.