In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.

中文翻译：

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不变离散流

在本文中，我们使用等变运动框架和诱导不变离散变分复合体理论研究了不变几何流下关节不变性的演化。对于某些保留弧长的平面曲线，在特殊欧几里得群，特殊线性群和半直群下不变，我们发现离散曲率的诱导演化分别满足微分差mKdV，KdV和Burgers方程。这三个方程是完全可积分的，并且我们表明可以通过用某个不变差分算子对曲率的特征算子进行预组合来构造递归算子。最后，我们推导出了离散曲率演化的可积性提升到离散曲线本身的演化的约束。