Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 032, 8 pages      arXiv:2201.11264

Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property

Nobutaka Nakazono
Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho Koganei, Tokyo 184-8588, Japan

Received February 03, 2022, in final form April 14, 2022; Published online April 20, 2022

The lattice sine-Gordon equation is an integrable partial difference equation on ${\mathbb Z}^2$, which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property as well as its autonomous version. Moreover, we construct two new Lax pairs of the non-autonomous case by using the consistency property.

Key words: lattice sine-Gordon equation; Lax pair; integrable systems; partial difference equations.

pdf (339 kb)   tex (32 kb)  


  1. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, arXiv:nlin.SI/0202024.
  2. Adler V.E., Bobenko A.I., Suris Yu.B., Discrete nonlinear hyperbolic equations: classification of integrable cases, Funct. Anal. Appl. 43 (2009), 3-17, arXiv:0705.1663.
  3. Bobenko A., Kutz N., Pinkall U., The discrete quantum pendulum, Phys. Lett. A 177 (1993), 399-404.
  4. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), 573-611, arXiv:nlin.SI/0110004.
  5. Boll R., Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 18 (2011), 337-365, arXiv:1009.4007.
  6. Boll R., Corrigendum: Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 19 (2012), 1292001, 3 pages.
  7. Capel H.W., Nijhoff F.W., Papageorgiou V.G., Complete integrability of Lagrangian mappings and lattices of KdV type, Phys. Lett. A 155 (1991), 377-387.
  8. Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2016.
  9. Joshi N., Nakazono N., On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries equation, Stud Appl. Math. 147 (2021), 1409-1424, arXiv:2102.00684.
  10. Kajiwara K., Ohta Y., Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation, J. Phys. Soc. Japan 77 (2008), 054004, 9 pages, arXiv:0802.0757.
  11. Kassotakis P., Nieszporski M., Difference systems in bond and face variables and non-potential versions of discrete integrable systems, J. Phys. A: Math. Theor. 51 (2018), 385203, 21 pages, arXiv:1710.11111.
  12. Nakazono N., Discrete Painlevé transcendent solutions to the multiplicative type discrete KdV equations, J. Math. Phys., to appear, arXiv:2104.11433.
  13. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, arXiv:nlin.SI/0110027.
  14. Nijhoff F.W., Capel H.W., Wiersma G.L., Quispel G.R.W., Bäcklund transformations and three-dimensional lattice equations, Phys. Lett. A 105 (1984), 267-272.
  15. Nijhoff F.W., Quispel G.R.W., Capel H.W., Direct linearization of nonlinear difference-difference equations, Phys. Lett. A 97 (1983), 125-128.
  16. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J. 43A (2001), 109-123, arXiv:nlin.SI/0001054.
  17. Nimmo J.J.C., Schief W.K., An integrable discretization of a $(2+1)$-dimensional sine-Gordon equation, Stud. Appl. Math. 100 (1998), 295-309.
  18. Quispel G.R.W., Nijhoff F.W., Capel H.W., van der Linden J., Linear integral equations and nonlinear difference-difference equations, Phys. A 125 (1984), 344-380.
  19. Tremblay S., Grammaticos B., Ramani A., Integrable lattice equations and their growth properties, Phys. Lett. A 278 (2001), 319-324, arXiv:0709.3095.
  20. Volkov A.Yu., Faddeev L.D., Quantum inverse scattering method on a spacetime lattice, Theoret. and Math. Phys. 92 (1992), 837-842.
  21. Walker A., Similarity reductions and integrable lattice equations, Ph.D. Thesis, University of Leeds, 2001.

Previous article  Next article  Contents of Volume 18 (2022)