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.
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References
M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics 4 (SIAM, Philadelphia, 1981)
A. Doliwa, P.M. Santini, Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy. J. Math. Phys. 36, 1259–1273 (1995)
B.F. Feng, J. Inoguchi, K. Kajiwara, K. Maruno, Y. Ohta, Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves. J. Phys. A Math. Theor. 44, 395201 (2011)
R.E. Goldstein, D.M. Petrich, The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane. Phys. Rev. Lett. 67, 3203–3206 (1991)
R. Hirota, Exact N-soliton solution of nonlinear lumped self-dual network equation. J. Phys. Soc. Jpn. 35, 289–294 (1973)
R. Hirota, Discretization of the potential modified KdV equation. J. Phys. Soc. Jpn. 67, 2234–2236 (1998)
T. Hoffmann, Discrete differential geometry of curves and surfaces, COE Lecture Notes, vol. 18 (Kyushu University, Fukuoka, 2009)
J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Motion and Bäcklund transformations of discrete plane curves. Kyushu J. Math. 66, 303–324 (2012)
J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves. J. Phys. A: Math. Theor. 45, 045206 (2012)
J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Discrete mKdV and discrete sine-Gordon flows on discrete space curves. J. Phys. A: Math. Theor. 47, 235202 (2014)
N. Matsuura, Discrete KdV and discrete modified KdV equations arising from motions of discrete planar curves. Int. Math. Res. Notices 2012, 1681–1698 (2012)
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Inoguchi, Ji., Kajiwara, K., Matsuura, N., Ohta, Y. (2014). Discrete Models of Isoperimetric Deformation of Plane Curves. In: Nishii, R., et al. A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_7
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DOI: https://doi.org/10.1007/978-4-431-55060-0_7
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