Discrete mKdV and discrete sine-Gordon flows on discrete space curves

Abstract

In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric condition. The curve is reconstructed from the deformation data by using the Sym-Tafel formula. The isoperimetric equidistant deformation of the space curves does not preserve the torsion in general. However, it is possible to construct the torsionpreserving deformation by tuning the deformation parameters. Further, it is also possible to make an arbitrary choice of the deformation described by the discrete mKdV equation or by the discrete sine-Gordon equation at each step. We finally show that the discrete deformation of discrete space curves yields the discrete K-surfaces.

Invariant discrete flows

“…We note that the equality (34) can be verified using the coordinate expressions (16) for the joint invariants. Therefore, the Maurer-Cartan invariants are…”

“…Remark 1. As the reader will have observed in Example 3, in (16) we considered the backward shift of ( , ) when invariantizing −1 and −1 in (16) and have not limited ourselves to forward shifts as in the above theoretical exposition. We did so to obtain formulas for the joint invariants that are more commonly found in the literature 29 .…”

“…Example 5. Continuing Example 3, we now derive the recurrence relations (23) for the normalized invariants (16). First, a basis of (prolonged) infinitesimal generators of the special Euclidean group action (13) is given by…”

“…Last, in Ref. it is shown that under a continuous isoperimetric deformation of a discrete curve in with constant torsion, the evolution of the discrete curvature is again given by the differential‐difference mKdV.…”

Nonlinear d’Alembert formula for discrete pseudospherical surfaces

“…In [14], a discrete analogue of the flow of constant torsion curves was introduced. In this appendix we show that the discrete flow of a discrete constant torsion curve can be identified with a discrete PS surface.…”