Discrete Models of Isoperimetric Deformation of Plane Curves
Discrete Models of Isoperimetric Deformation of Plane Curves
复制标题
平面曲线等周变形的离散模型
DOI：
10.1007/978-4-431-55060-0_7
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Nozomu Matsuura and Yasuhiro Ohta
中科院分区：
文献类型：
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作者：
Jun-ichi Inoguchi;Kenji Kajiwara;Nozomu Matsuura and Yasuhiro Ohta
我们考虑欧氏平面上光滑曲线的等周变形。它自然而然地产生了一个被称为修正的KdV(MKdV)方程的非线性偏微分方程曲率的变形方程，它被称为孤子方程或可积系统的最典型的例子之一。曲线的Frenet方程和Frenet标架的变形方程是mKdV方程的辅助线性问题或Lax对。在此基础上，我们提出了两种保持基本可积结构的平面曲线等周变形的离散模型：由离散mKdV方程描述的离散变形和由半离散mKdV方程描述的连续变形。
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.