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Equivalent elastica knots
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The problem of an elastica knot in three-dimensional space is solved explicitly by expressing the Frenet-Serret curvature and torsion of the knot in terms of the Weierstrass and Jacobi elliptic functions. This solution is obtained by variational methods and is derived by minimizing of the squared-curvature energy integral. In the present work, an equivalency is established between pairs of Jacobi elliptic solutions that are described by the same values for curvature and torsion functionals.
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Nonlinear Schr\"{o}dinger equation on a closed 3D elastica knot
Real-speed NLSE traveling waves can match closed elastica knots only when the elliptic parameter m is below about -4.75, outside the classical elastica-knot range 0 ≤ m ≤ 1.
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