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Classification of Three-Dimensional Integrable Scalar Discrete Equations

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Abstract

We classify all integrable three-dimensional scalar discrete affine linear equations Q 3 = 0 on an elementary cubic cell of the lattice \({\mathbb Z}^3\) . An equation Q 3 = 0 is called integrable if it may be consistently imposed on all three-dimensional elementary faces of the lattice \({\mathbb Z}^4\) . Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only non-trivial (non-linearizable) integrable equation from this class is the well-known dBKP-system.

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References

  • Adler V.E., Bobenko A.I. and Suris Yu.B. (2003). Classification of integrable equations on quad-graphs. The consistency approach. Commun. Math. Phys. 233: 513–543

    MATH  ADS  MathSciNet  Google Scholar 

  • Adler, V.E., Bobenko, A.I., Suris, Yu.B.: Discrete nonlinear hyperbolic equations. Classification of integrable cases. Preprint at http://www.arxiv.org, nlin.SI/ 0705.1663

  • Bobenko, A.I., Suris, Yu.B.: Discrete differential geometry. Consistency as integrability. Preliminary version of a book. Preprint at http://www.arxiv.org, math.DG/0504358 (2005)

  • Bobenko, A.I., Suris, Yu.B.: On organizing principles of Discrete Differential Geometry. Geometry of spheres. Russian Math. Surv. 62(1), 1–43 (2007). Also http://www.arXiv.org, math/0608291

  • Ganzha, E.I., Tsarev, S.P.: On superposition of the auto-Baecklund transformations for (2+1)-dimensional integrable systems. Russian Math. Surv. 51(6), 1200–1202 (1996). See also http://www.arxiv.org, solv-int/9606003

  • Konopelchenko, B.G., Schief, W.K.: Reciprocal figures, graphical statics and inversive geometry of the Schwarzian BKP hierarchy. Stud. Appl. Math. 109, 89–124 (2002). Also preprint nlin.SI/0107001 at http://www.arXiv.org

  • Nimmo J.J.C. and Schief W.K. (1998). an integrable discretization of a 2 + 1-dimensional sine-Gordon equation. Stud. Appl. Math. 100: 295–309

    Article  MATH  MathSciNet  Google Scholar 

  • Papageorgiou, V.G., Tongas, A.G., Veselov, A.P.: Yang–Baxter maps and symmetries of integrable equations on quad-graphs. J. Math. Phys. 47(8), No. 083502, 16 pp (2006)

    Google Scholar 

  • Liu, Y., Pottmann, H., Wallner, J., Yang, Y., Wang, W.: Geometric modelling with conical meshes and developpable surfaces. ACM Trans. Graph. 25(3), 681–689 (2006), Proc. SIGGRAPH 2006. See also http://dmg.tuwien.ac.at/pottmann/

    Google Scholar 

  • Vermaseren, J.A.M.: New features of FORM. arXiv:math-ph/0010025, a complete distribution can be downloaded from http://www.nikhef.nl/~form/

  • Wolf, T.: Applications of CRACK in the classification of integrable systems. In: CRM Proceedings and Lecture Notes, vol. 37, pp. 283–300 (2004). Available http://www.arxiv.org, nlin.SI/0301032

  • Wolf, T.: An Online Tutorial for the package Crack. http://lie.math.brocku.ca/crack/demo

  • Tsarev, S.P., Wolf, T.: Classification of 3-dimensional integrable scalar discrete equations. Preprint at http://www.arxiv.org, nlin.SI/0706.2464 (2007). The commented Form codes related to this publication can be downloaded from http://lie.math.brocku.ca/twolf/papers/TsWo2007/

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Correspondence to Thomas Wolf.

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Supported by the DFG Research Unit 565 “Polyhedral Surfaces” (TU-Berlin).

SPT acknowledges partial financial support from a grant of Siberian Federal University (NM-project No 45.2007) and the RFBR grant 06-01-00814.

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Tsarev, S.P., Wolf, T. Classification of Three-Dimensional Integrable Scalar Discrete Equations. Lett Math Phys 84, 31–39 (2008). https://doi.org/10.1007/s11005-008-0230-2

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  • DOI: https://doi.org/10.1007/s11005-008-0230-2

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