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On Generalizations of the Pentagram Map: Discretizations of AGD Flows

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Abstract

In this paper we investigate discretizations of AGD flows whose projective realizations are defined by intersecting different types of subspace in \(\mathbb{RP}^{m}\). These maps are natural candidates to generalize the pentagram map, itself defined as the intersection of consecutive shortest diagonals of a convex polygon, and a completely integrable discretization of the Boussinesq equation. We conjecture that the r-AGD flow in m dimensions can be discretized using one (r−1)-dimensional subspace and r−1 different (m−1)-dimensional subspaces of \(\mathbb{RP}^{m}\).

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References

  • Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV. Invent. Math. 50, 219–248 (1979)

    Article  MATH  Google Scholar 

  • Calini, A., Ivey, T., Marí Beffa, G.: An integrable flow for starlike curves in centroaffine space (2012, submitted)

  • Dickson, R., Gesztesy, F., Unterkoer, K.: Algebro-geometric solutions of the Boussinesq hierarchy. Rev. Math. Phys. 11, 823–879 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Drinfel’d, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg–de Vries type. In: Current Problems in Mathematics. Itogi Nauki i Tekhniki, vol. 24, pp. 81–180. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform, Moscow (1984)

    Google Scholar 

  • Fels, M., Olver, P.J.: Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math., 127–208 (1999)

  • Gel’fand, I.M., Dickey, L.A.: A family of Hamiltonian structures connected with integrable nonlinear differential equations. In: Gelfand, I.M. (ed.) Collected Papers, vol. 1. Springer, Berlin (1987)

    Chapter  Google Scholar 

  • Gekhtman, M., Shapiro, M., Tabachnikov, S., Vainshtein, A.: Higher pentagram maps, weighted directed networks, and cluster dynamics. Electron. Res. Announc. Math. Sci. 19, 1–17 (2012)

    MathSciNet  MATH  Google Scholar 

  • Heredero, R., Lopez, A., Marí Beffa, G.: Invariant differential equations and the Adler–Gel’fand–Dikii bracket. J. Math. Phys. 38, 5720–5738 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Hubert, E.: Generation properties of differential invariants in the moving frame methods. Preprint http://hal.inria.fr/inria-00194528/en (2007)

  • Khesin, B., Soloviev, F.: Integrability of higher pentagram maps (2012a). arXiv:1204.0756

  • Khesin, B., Soloviev, F.: The pentagram map in higher dimensions and KdV flows (2012b). arXiv:1205.3744

  • Lobb, S.B., Nijhoff, F.W.: Lagrangian multiform structure for the lattice Gel’fand–Dikii hierarchy. J. Phys. A 43(7) (2010)

  • Marí Beffa, G.: The theory of differential invariants and KdV Hamiltonian evolutions. Bull. Soc. Math. Fr. 127, 363–391 (1999)

    MATH  Google Scholar 

  • Marí Beffa, G.: Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces. Proc. Am. Math. Soc. 134, 779–791 (2006)

    Article  MATH  Google Scholar 

  • Marí Beffa, G.: Geometric Hamiltonian structures on flat semisimple homogeneous manifolds, the Asian. J. Math. 12(1), 1–33 (2008)

    MATH  Google Scholar 

  • Marí Beffa, G.: On bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds. Pac. J. Math. 247(1), 163–188 (2010)

    Article  MATH  Google Scholar 

  • Ovsienko, V., Schwartz, R., Tabachnikov, S.: The pentagram map: a discrete integrable system. Commun. Math. Phys. 299, 409–446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Ovsienko, V., Schwartz, R., Tabachnikov, S.: Liouville–Arnold Integrability of the pentagram map on closed polygons (2011). arXiv:1107.3633

  • Schwartz, R.: The pentagram map is recurrent. J. Exp. Math. 10.4, 519–528 (2001)

    Article  Google Scholar 

  • Schwartz, R.: Discrete monodromy, pentagrams and the method of condensation. J. Fixed Point Theory Appl. 3, 379–409 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Schwartz, R., Tabachnikov, S.: The pentagram integrals for inscribed polygons. Electron. J. Combin. 18(1), 171 (2011)

    MathSciNet  Google Scholar 

  • Soloviev, F.: Integrability of the pentagram map (2011). arXiv:1106.3950

  • Wilczynski, E.J.: Projective Differential Geometry of Curves and Ruled Surfaces. Teubner, Leipzig (1906)

    MATH  Google Scholar 

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Acknowledgement

This paper is supported by the author NSF grant DMS #0804541.

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  1. University of Wisconsin, Madison, USA

    Gloria Marí Beffa

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  1. Gloria Marí Beffa
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Correspondence to Gloria Marí Beffa.

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Communicated by A. Block.

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Marí Beffa, G. On Generalizations of the Pentagram Map: Discretizations of AGD Flows. J Nonlinear Sci 23, 303–334 (2013). https://doi.org/10.1007/s00332-012-9152-3

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