The pentagram map in higher dimensions and KdV flows

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2012, 19: 86-96. doi: 10.3934/era.2012.19.86

The pentagram map in higher dimensions and KdV flows

Boris Khesin ^1, and Fedor Soloviev ^2,

1.	School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA and Department of Mathematics,, University of Toronto, Toronto, ON M5S 2E4, Canada
2.	Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Published September 2012

Abstract
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We extend the definition of the pentagram map from 2D to higher dimensions and describe its integrability properties for both closed and twisted polygons by presenting its Lax form. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D.

Keywords: Lax equation, pentagram map, KdV hierarchy, completely integrable system, spectral curve..

Mathematics Subject Classification: Primary: 37J35; Secondary: 37K10, 37K25, 53A2.

Citation: Boris Khesin, Fedor Soloviev. The pentagram map in higher dimensions and KdV flows. Electronic Research Announcements, 2012, 19: 86-96. doi: 10.3934/era.2012.19.86

References:

[1]	M. Gekhtman, M. Shapiro, S. Tabachnikov and A. Vainshtein, Higher pentagram maps, weighted directed networks, and cluster dynamics, Electron. Res. Announc. Math. Sci., 19 (2012), 1-17; arXiv:1110.0472.
[2]	B. Khesin and F. Soloviev, Integrability of higher pentagram maps, (2012); arXiv:1204.0756.
[3]	I. M. Krichever and D. H. Phong, On the integrable geometry of soliton equations and N=2 supersymmetric gauge theories, J. Diff. Geom., 45 (1997), 349-389.
[4]	I. M. Krichever and D. H. Phong, Symplectic forms in the theory of solitons, in "Surveys in Differential Geometry: Integral Systems [Integrable Systems]," Surv. Diff. Geom., Vol. IV, Int. Press, Boston, MA, (1998), 239-313.
[5]	G. Marí Beffa, On generalizations of the pentagram map: Discretizations of AGD flows, (2011); arXiv:1103.5047.
[6]	V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system, Comm. Math. Phys., 299 (2010), 409-446; arXiv:0810.5605. doi: 10.1007/s00220-010-1075-y.
[7]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, (2011); arXiv:1107.3633.
[8]	R. Schwartz, The pentagram map, Experiment. Math., 1 (1992), 71-81.
[9]	F. Soloviev, Integrability of the pentagram map, submitted to Duke Mathematical Journal, (2011); arXiv:1106.3950.

show all references

References:

[1]	M. Gekhtman, M. Shapiro, S. Tabachnikov and A. Vainshtein, Higher pentagram maps, weighted directed networks, and cluster dynamics, Electron. Res. Announc. Math. Sci., 19 (2012), 1-17; arXiv:1110.0472.
[2]	B. Khesin and F. Soloviev, Integrability of higher pentagram maps, (2012); arXiv:1204.0756.
[3]	I. M. Krichever and D. H. Phong, On the integrable geometry of soliton equations and N=2 supersymmetric gauge theories, J. Diff. Geom., 45 (1997), 349-389.
[4]	I. M. Krichever and D. H. Phong, Symplectic forms in the theory of solitons, in "Surveys in Differential Geometry: Integral Systems [Integrable Systems]," Surv. Diff. Geom., Vol. IV, Int. Press, Boston, MA, (1998), 239-313.
[5]	G. Marí Beffa, On generalizations of the pentagram map: Discretizations of AGD flows, (2011); arXiv:1103.5047.
[6]	V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system, Comm. Math. Phys., 299 (2010), 409-446; arXiv:0810.5605. doi: 10.1007/s00220-010-1075-y.
[7]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, (2011); arXiv:1107.3633.
[8]	R. Schwartz, The pentagram map, Experiment. Math., 1 (1992), 71-81.
[9]	F. Soloviev, Integrability of the pentagram map, submitted to Duke Mathematical Journal, (2011); arXiv:1106.3950.

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