Matsutani, Shigeki; Previato, Emma From Euler’s elastica to the mKdV hierarchy, through the Faber polynomials. (English) Zbl 1398.37065 J. Math. Phys. 57, No. 8, 081519, 12 p. (2016). Summary: The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler’s elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.{©2016 American Institute of Physics} Cited in 3 Documents MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q53 KdV equations (Korteweg-de Vries equations) 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 11B83 Special sequences and polynomials Keywords:Korteweg-de Vries hierarchy; isometry/isoenergy conditions; Euler’s elastica PDFBibTeX XMLCite \textit{S. Matsutani} and \textit{E. Previato}, J. Math. Phys. 57, No. 8, 081519, 12 p. (2016; Zbl 1398.37065) Full Text: DOI arXiv References: [1] Anco, S. C.; Myrzakulov, R., Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces, J. Geom. Phys., 60, 10, 1576-1603 (2010) · Zbl 1195.53062 [2] Bott, R.; Tu, L. W., Differential Forms in Algebraic Topology, 82 (1982) · Zbl 0496.55001 [3] Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, 107 (1993) · Zbl 0823.55002 [4] Ford, D. J.; McKay, J.; Norton, S. P., More on replicable functions, Commun. Algebra, 22, 5175-5193 (1994) · Zbl 0834.11021 [5] Goldstein, R. E.; Petrich, D. M., The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett., 67, 3203-3206 (1991) · Zbl 0990.37519 [6] Goldstein, R. E.; Petrich, D. M., Solitons, Euler’s equation, and the geometry of curve motion, Singularities in Fluids, Plasmas and Optics, Heraklion, 1992, 404, 93-109 (1993) [7] Levien, R., The elastica: A mathematical history, 2008, . [8] Levien, R. L., From spiral to spline: Optimal techniques in interactive curve design (2009) [9] Matsutani, S., Statistical mechanics of elastica on a plane: Origin of mKdV hierarchy, J. Phys. A, 31, 2705-2725 (1998) · Zbl 0919.35121 [10] Matsutani, S., Hyperelliptic loop solitons with genus g: Investigations of a quantized elastica, J. Geom. Phys., 43, 146-162 (2002) · Zbl 0998.35045 [11] Matsutani, S., On the moduli of a quantized elastica in ℙ and KdV flows: Study of hyperelliptic curves as an extension of Euler’s perspective of elastica I, Rev. Math. Phys., 15, 559-628 (2003) · Zbl 1079.58006 [12] Matsutani, S., Relations in a quantized elastica, J. Phys. A: Math. Theor., 41, 075201 (2008) · Zbl 1134.37035 [13] Matsutani, S., Euler’s elastica and beyond, J. Geom. Symmetry Phys., 17, 45-86 (2010) · Zbl 1209.37086 [14] Matsutani, S. and Previato, E., “Algebraic and analytic identities for the Faber polynomials,” preprint (2016). · Zbl 1398.37065 [15] Miura, R. M., Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., 9, 1202-1204 (1968) · Zbl 0283.35018 [16] Saitô, N.; Takaiashi, K.; Yunoki, Y., The statistical mechanics theory of stiff chains, J. Phys. Soc. Jpn., 22, 1, 219-226 (1967) [17] Tjurin, A. N., Periods of quadratic differentials, Usp. Mat. Nauk, 33, 6, 149-195 (1978) · Zbl 0413.30038 [18] Truesdell, C., The influence of elasticity on analysis: The classic heritage, Bull. Am. Math. Soc., 9, 293-310 (1983) · Zbl 0555.73030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.