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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(2), the special linear group SL(2), and the semidirect group RR2, 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 curveĀ itself.
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Keywords: discrete geometric flows; discrete variational complex; equivariant moving frames; joint invariants; lie group actions

Document Type: Research Article

Publication date: July 1, 2019

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