Abstract
The new idea of flip invariance of action functionals in multidimensional lattices was recently highlighted as a key feature of discrete integrable systems. Flip invariance was proved for several particular cases of integrable quad-equations by Bazhanov, Mangazeev and Sergeev and by Lobb and Nijhoff. We provide a simple and case-independent proof for all integrable quad-equations. Moreover, we find a new relation for Lagrangians within one elementary quadrilateral which seems to be a fundamental building block of the various versions of flip invariance.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler V.E., Bobenko A.I., Suris Yu.B.: Classification of integrable equations on quad-graphs. The consistency approach. Commun. Math. Phys. 233, 513–543 (2003)
Adler V.E., Bobenko A.I., Suris Yu.B.: Discrete nonlinear hyperbolic equations. Classification of integrable cases. Funct. Anal. Appl. 43, 3–17 (2009)
Adler V.E., Suris Yu.B.: Q4: integrable master equation related to an elliptic curve. Int. Math. Res. Notice 47, 2523–2553 (2004)
Baxter R.J.: Exactly Solved Models in Statistical Mechanics, xii+486 pp. Academic Press, London (1982)
Bazhanov V.V., Mangazeev V.V., Sergeev S.M.: Faddeev–Volkov solution of the Yang–Baxter equation and discrete conformal geometry. Nucl. Phys. B 784, 234–258 (2007)
Bazhanov V.V., Mangazeev V.V., Sergeev S.M.: Quantum geometry of 3-dimensional lattices. J. Stat. Mech. 0807, P07004 (2008)
Bobenko A.I., Springborn B.A.: Variational principles for circle patterns and Koebe’s theorem. Trans. Am. Math. Soc. 356, 659–689 (2004)
Bobenko A.I., Suris Yu.B.: Integrable systems on quad-graphs. Int. Math. Res. Notices 11, 573–611 (2002)
Bobenko, A.I., Suris, Yu.B.: Discrete differential geometry: integrable structures. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence (2008)
Faddeev L.D., Volkov A.Yu.: Abelian current algebra and the Virasoro algebra on the lattice. Phys. Lett. B 315, 311–318 (1993)
Lobb S., Nijhoff F.W. Lagrangian multiforms and multidimensional consistency. J. Phys. A: Math. Theor. 42(45), paper id. 454013, 18 pp (2009)
Lobb S.B., Nijhoff F.W., Quispel, G.R.W.: Lagrangian multiform structure for the lattice KP system. J. Phys. A: Math. Theor. 42(47), paper id. 472002, 11 pp (2009)
Marsden J.E., Patrick G.W., Shkoller S.: Mulltisymplectic geometry, variational integrators and nonlinear PDEs. Commun. Math. Phys. 199, 351–395 (1998)
Moser J., Veselov A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217–243 (1991)
Nijhoff F.W.: Lax pair for the Adler (lattice Krichever–Novikov) system. Phys. Lett. A 297(1–2), 49–58 (2002)
Suris, Yu.B.: Discrete Lagrangian models. In: Grammaticos, B., Kosmann- Schwarzbach, Y., Tamizhmani, T. (eds.) Discrete Integrable Systems. Lecture Notes Phys., vol. 644, pp. 111–184. Springer, Berlin (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
A.I. Bobenko was partially supported by the DFG Research Unit “Polyhedral Surfaces”.
Rights and permissions
About this article
Cite this article
Bobenko, A.I., Suris, Y.B. On the Lagrangian Structure of Integrable Quad-Equations. Lett Math Phys 92, 17–31 (2010). https://doi.org/10.1007/s11005-010-0381-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-010-0381-9