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

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.

19 pages

References in corpus (2)

• The short pulse equation is integrable
• An integrable semi-discretization of the Camassa-Holm equation and its determinant solution

Cited by in corpus (4)

• Multi-soliton, multi-breather and higher-order rogue wave solutions to the complex short pulse equation
• Complex short pulse and coupled complex short pulse equations
• Integrable semi-discretization of a multi-component short pulse equation
• $N$-soliton formula and blowup result of the Wadati-Konno-Ichikawa equation