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

TL;DR

This paper develops discrete analogues of hodograph transformations to create integrable discretizations of soliton equations linked to plane curve motions, enhancing the understanding of discrete integrable systems.

Contribution

It introduces discrete hodograph transformations as Euler-Lagrange transformations, providing new integrable discretizations of soliton equations related to plane curve motions.

Findings

Constructed discrete hodograph transformations for integrable discretizations.

Connected discrete curve motions to well-known soliton equations.

Established a framework for discretizing curve-related integrable systems.

Abstract

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.