Pinkall在1995年的論文中推倒了中心仿射平面閉星形曲線空間上的
哈密頓演化方程，以及證明了中心仿射曲線流和KdV 方程有著密切的相
關。到了2013年 Terng, Chuu-Lian 和 Zhiwei Wu 在他們的論文中研
究中心仿射曲線流的週期性初始值問題以及提出了數值解的演算方
法。與中心仿射曲線流有著密切相關的KdV方程為著名的弱非線性偏微
分方程，在數學和物理界有著深遠的影響，此方程觀察了淺水中波的傳
遞，而孤立子為KdV方程的其中一解，孤立子有著穩定傳遞與自我保持
的特性，因此被廣泛應用在各種物理系統中。
  此篇論文研究了與KdV方程有著密切相關的中心仿射曲線流的數值
解，為了觀察(週期性)柯西問題的長時間行為，我們應用了 Palais 的
WGMS方法來解KdV 方程的週期解，接著用中心仿射曲線流與KdV 方程之
間的關係來做數值解的演算法。在此篇論文的文末我們用中心仿射曲線
流的穩定解(圓和橢圓)來估計真實解與數值解的誤差，並且我們也展示
了以五星形為初始值曲線的時間變化圖以及初始值曲線為與五星形曲
線相似的圖形的例子。

  In 1995, Pinkall derived the Hamiltonian evolution equation on the space
of closed affine plane star-shaped curves, establishing the close
relationship between affine curve flows and the KdV equation. By 2013, Terng,
Chuu-Lian, and Zhiwei Wu studied the periodic initial value problem for affine
curve flows and proposed numerical algorithms for solving it in their paper.
The KdV equation, closely associated with central affine curve flows, is a
well-known weakly nonlinear partial differential equation with profound
implications in mathematics and physics. It describes the propagation of waves
in shallow water, with solitons being particular solutions known for their
stable propagation and self-preservation properties, widely applicable in
various physical systems.
  We aim to study the numerical solution of the central affine curve flow,
which has the close relationship with the well-known KdV equation. In order
to observe the long-time behavior of periodic Cauchy problems, we applied
Palais's WGMS approach to solve periodic solutions of the KdV equation,
followed by developing numerical algorithms based on the relationship between
affine curve flows and the KdV equation. At the conclusion of this paper,
stable solutions of affine curve flows (such as the circle and the ellipse)
were used to estimate errors between real solutions and numerical solutions,
accompanied by time evolution plots of star-shaped curve and similar figures.

1. Introduction   2
2. Relation of the KdV equation and the central affine curve flow on R2   3
3. Algorithm of the central affine curve flow   5
4. Error estimates   8
5. Experimental issues and Examples of numerical solutions   12
6. Experimental codes  15
References   19

1. Terng, Chuu-Lian and Zhiwei Wu, “Central affine curve flow on the plane”, Journal of Fixed Point Theory and Applications 14 (2013): 375-396.
2. Pinkall, Ulrich, “Hamiltonian flows on the space of star-shaped curves”, Results in Mathematics 27 (1995): 328-332.
3. Palais, Richard, “The initial value problem for weakly nonlinear PDE”, Journal of Fixed Point Theory and Applications 16 (2014): 337-349.
4. Schalch, Nicolas, “The Korteweg-de Vries Equation”, (2018).
5. Crighton, Diane, “Applications of KdV”, Acta Applicandae Mathematica 39 (1995): 39-67.
6. Calini, Annalisa, Thomas A. Ivey and Gloria Mar ́ı Beffa, “Remarks on KdV-type flows on star-shaped
curves”, Physica D: Nonlinear Phenomena 238 (2008): 788-797.