S. Konstantinou-Rizos and T.E. Kouloukas, "A noncommutative discrete potential KdV lift", Journal of Mathematical Physics, 59, 063506 (2018).

S. Konstantinou-Rizos and T.E. Kouloukas, A noncommutative discrete potential KdV lift, Journal of Mathematical Physics, 59, 063506 (2018).



In this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared in the work of Kouloukas and Papageorgiou [J. Phys. A: Math. Theor. 42, 404012 (2009)] and can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension satisfies the Yang-Baxter equation, and it admits a 3 × 3 Lax matrix. Moreover, we show that it can be squeezed down to a novel system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, we consider commutative analogs of the constructed Yang-Baxter map and its associated quad-graph system, and we discuss their integrability.


  1. 1.Adler, V., Bobenko, A., and Suris, Y., “Classification of integrable equations on quad-graphs. The consistency approach,” Commun. Math. Phys. 233, 513–543 (2003). https://doi.org/10.1007/s00220-002-0762-8Google ScholarCrossref
  2. 2.Adler, V., Bobenko, A., and Suris, Y., “Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings,” Commun. Anal. Geom. 12, 967–1007 (2004). https://doi.org/10.4310/cag.2004.v12.n5.a1Google ScholarCrossref
  3. 3.Berezin, F., Intoduction to Superanalysis (D. Reidel, Dordrecht, 1987). Google ScholarCrossref
  4. 4.Bobenko, A. and Suris, Y., “Integrable systems on quad-graphs,” Int. Math. Res. Not. 11, 573–611 (2002). https://doi.org/10.1155/s1073792802110075Google ScholarCrossref
  5. 5.Bridgman, T., Hereman, W., Quispel, G. R. W., and van der Kamp, P. H., “Symbolic computation of Lax pairs of partial difference equations using consistency around the cube,” Found. Comput. Math. 13, 517–544 (2013). https://doi.org/10.1007/s10208-012-9133-9Google ScholarCrossref
  6. 6.Buchstaber, V., “The Yang-Baxter transformation,” Russ. Math. Surv. 53, 1343–1345 (1998). https://doi.org/10.1070/rm1998v053n06abeh000094Google ScholarCrossref
  7. 7.Doliwa, A., “Non-commutative rational Yang-Baxter maps,” Lett. Math. Phys. 104, 299–309 (2014). https://doi.org/10.1007/s11005-013-0669-7Google ScholarCrossref
  8. 8.Drinfeld, V., “On some unsolved problems in quantum group theory,” Lect. Notes Math. 1510, 1–8 (1992). https://doi.org/10.1007/bfb0101175Google ScholarCrossref
  9. 9.Etingof, P., Schedler, T., and Soloviev, A., “Set-theoretical solutions to the quantum Yang-Baxter equation,” Duke Math. J. 100, 169–209 (1999). https://doi.org/10.1215/s0012-7094-99-10007-xGoogle ScholarCrossref
  10. 10.Grahovski, G., Konstantinou-Rizos, S., and Mikhailov, A., “Grassmann extensions of Yang-Baxter maps,” J. Phys. A: Math. Theor. 49, 145202 (2016). https://doi.org/10.1088/1751-8113/49/14/145202Google ScholarCrossref
  11. 11.Grahovski, G. and Mikhailov, A., “Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras,” Phys. Lett. A 377, 3254–3259 (2013). https://doi.org/10.1016/j.physleta.2013.10.018Google ScholarCrossref
  12. 12.Hirota, R., “Nonlinear partial difference equations. I. A difference analog of the Korteweg-de Vries equation,” J. Phys. Soc. Jpn. 43, 1423–1433 (1977). https://doi.org/10.1143/jpsj.43.1424Google ScholarCrossref
  13. 13.Kassotakis, P. and Nieszporski, M., “On non-multiaffine consistent-around-the-cube lattice equations,” Phys. Lett. A 376, 3135–3140 (2012). https://doi.org/10.1016/j.physleta.2012.10.009Google ScholarCrossref
  14. 14.Konstantinou-Rizos, S. and Mikhailov, A., “Darboux transformations, finite reduction groups and related Yang-Baxter maps,” J. Phys. A: Math. Theor. 46, 425201 (2013). https://doi.org/10.1088/1751-8113/46/42/425201Google ScholarCrossref
  15. 15.Konstantinou-Rizos, S. and Mikhailov, A., “Anticommutative extension of the Adler map,” J. Phys. A: Math. Theor. 49, 30LT03 (2016). https://doi.org/10.1088/1751-8113/49/30/30lt03Google ScholarCrossref
  16. 16.Kouloukas, T. E. and Papageorgiou, V. G., “Yang–Baxter maps with first-degree-polynomial 2 × 2 Lax matrices,” J. Phys. A: Math. Theor. 42, 404012 (2009). https://doi.org/10.1088/1751-8113/42/40/404012Google ScholarCrossref
  17. 17.Kouloukas, T. E. and Papageorgiou, V. G., “Entwining Yang-Baxter maps and integrable lattices,” Banach Center Publ. 93, 163–175 (2011). https://doi.org/10.4064/bc93-0-13Google ScholarCrossref
  18. 18.Kouloukas, T. E. and Papageorgiou, V. G., “Poisson Yang-Baxter maps with binomial Lax matrices,” J. Math. Phys. 52, 073502 (2011). https://doi.org/10.1063/1.3601520Google ScholarScitationISI
  19. 19.Kouloukas, T. E. and Tran, D., “Poisson structures for lifts and periodic reductions of integrable lattice equations,” J. Phys. A: Math. Theor. 48, 075202 (2015). https://doi.org/10.1088/1751-8113/48/7/075202Google ScholarCrossref
  20. 20.Mikhailov, A., Papamikos, G., and Wang, J. P. “Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere,” Lett. Math. Phys. 106, 973–996(2016). https://doi.org/10.1007/s11005-016-0855-5Google ScholarCrossref
  21. 21.Nijhoff, F., Papageorgiou, V. G., Capel, H. W., and Quispel, G. R. W., “The lattice Gelfand-Dikii hierarchy,” Inv. Probl. 8, 597–621(1992). https://doi.org/10.1088/0266-5611/8/4/010Google ScholarCrossref
  22. 22.Nijhoff, F. and Capel, H., “The discrete Korteweg-de Vries equation,” Acta Appl. Math. 39, 133–158 (1995). https://doi.org/10.1007/bf00994631Google ScholarCrossref
  23. 23.Nijhoff, F. and Walker, A., “The discrete and continuous Painlevé VI hierarchy and the Garnier systems,” Glasgow Math. J. 43, 109–123 (2001). https://doi.org/10.1017/s0017089501000106Google ScholarCrossref
  24. 24.Papageorgiou, V. G., Nijhoff, F. W., and Capel, H. W., “Integrable mappings and nonlinear integrable lattice equations,” Phys. Lett. A 147, 106–114 (1990). https://doi.org/10.1016/0375-9601(90)90876-pGoogle ScholarCrossref
  25. 25.Papageorgiou, V. G. and Tongas, A. G., “Yang–Baxter maps and multi-field integrable lattice equations,” J. Phys. A: Math. Theor. 40, 12677 (2007). https://doi.org/10.1088/1751-8113/40/42/s12Google ScholarCrossref
  26. 26.Papageorgiou, V. G. and Tongas, A. G., “Yang-Baxter maps associated to elliptic curves,” e-print arXiv:0906.3258v1 (2009). Google Scholar
  27. 27.Papageorgiou, V. G., Tongas, A. G., and Veselov, A. P., “Yang–Baxter maps and symmetries of integrable equations on quad-graphs,” J. Math. Phys. 47, 083502 (2006). https://doi.org/10.1063/1.2227641Google ScholarScitationISI
  28. 28.Sklyanin, E., “Some algebraic structures connected with the Yang-Baxter equation,” Funct. Anal. Appl. 16, 263–270 (1982). https://doi.org/10.1007/bf01077848Google ScholarCrossref
  29. 29.Sklyanin, E., “Classical limits of SU(2)-invariant solutions of the Yang-Baxter equation,” J. Sov. Math. 40, 93–107 (1988). https://doi.org/10.1007/bf01084941Google ScholarCrossref
  30. 30.Sklyanin, E., “Separation of variables, new trends,” Prog. Theor. Phys. Suppl. 118, 35–60 (1995). https://doi.org/10.1143/ptps.118.35Google ScholarCrossref
  31. 31.Suris, Y. and Veselov, A., “Lax matrices for Yang-Baxter maps,” J. Nonlinear Math. Phys. 10, 223–230 (2003). https://doi.org/10.2991/jnmp.2003.10.s2.18Google ScholarCrossref
  32. 32.Veselov, A., “Yang-Baxter maps and integrable dynamics,” Phys. Lett. A 314, 214–221 (2003). https://doi.org/10.1016/s0375-9601(03)00915-0Google ScholarCrossrefCAS
  33. 33.Veselov, A., “Yang-Baxter maps: Dynamical point of view,” Math. Soc. Jpn. Mem. 17, 145–167 (2007). https://doi.org/10.2969/msjmemoirs/01701C060Google ScholarCrossref
  34. 34.Xue, L. L., Levi, D., and Liu, Q. P., “Supersymmetric KdV equation: Darboux transformation and discrete systems,” J. Phys. A: Math. Theor. 46, 502001 (2013). https://doi.org/10.1088/1751-8113/46/50/502001Google ScholarCrossref
  35. 35.Xue, L. L. and Liu, Q. P., “Bäcklund–Darboux transformations and discretizations of super KdV equation,” SIGMA 10, 045 (2014). https://doi.org/10.3842/SIGMA.2014.045Google ScholarCrossref
  36. 36.Xue, L. L. and Liu, Q. P., “A supersymmetric AKNS problem and its Darboux-Bäcklund transformations and discrete systems,” Stud. Appl. Math. 135, 35–62 (2015). https://doi.org/10.1111/sapm.12080Google ScholarCrossref