# Invariant discrete flows

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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 $SE\left(2\right)$, the special linear group $SL\left(2\right)$, and the semidirect group $\mathbb{R}⋉{\mathbb{R}}^{2}$, 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.
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Document Type: Research Article

Publication date: July 1, 2019

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