Discrete Models of Isoperimetric Deformation of Plane Curves

  • Jun-ichi Inoguchi
  • Kenji Kajiwara
  • Nozomu Matsuura
  • Yasuhiro Ohta
Chapter
Part of the Mathematics for Industry book series (MFI, volume 5)

Abstract

We consider the isoperimetric deformation of smooth curves on the Euclidean plane. It naturally gives rise to a nonlinear partial differential equation called the modified KdV(mKdV) equation as a deformation equation of the curvature, which is known as one of the most typical example of the soliton equations or the integrable systems. The Frenet equation and the deformation equation of the Frenet frame of the curve are the auxiliary linear problem or the Lax pair of the mKdV equation. Based on this formulation, we present two discrete models of isoperimetric deformation of plane curves preserving underlying integrable structure: the discrete deformation described by the discrete mKdV equation and the continuous deformation described by the semi-discrete mKdV equation.

Keywords

Plane curve Discrete plane curve Isoperimetric deformation mKdV equation Semi-discrete mKdV equation Discrete mKdV equation 

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Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Jun-ichi Inoguchi
    • 1
  • Kenji Kajiwara
    • 2
  • Nozomu Matsuura
    • 3
  • Yasuhiro Ohta
    • 4
  1. 1.Department of Mathematical SciencesYamagata UniversityYamagataJapan
  2. 2.Institute of Mathematics for IndustryFukuokaJapan
  3. 3.Department of Applied MathematicsFukuoka UniversityFukuokaJapan
  4. 4.Department of MathematicsKobe UniversityRokko, KobeJapan

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