Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.

💡 Research Summary

The paper investigates integrable discretizations of several soliton equations that arise from the motion of plane curves: the Wadati‑Konno‑Ichikawa (WKI) elastic beam equation, the complex Dym equation, and the short‑pulse equation. The authors start from the well‑known geometric description of curve evolution in the Euclidean plane. By parametrising a smooth curve γ(s,t) with arc‑length s, they introduce the Frenet frame (T,N) and the curvature κ=θ_s, where θ(s,t) is the turning angle. The isoperimetric flow leads to the potential modified KdV (mKdV) equation \