A noncommutative discrete potential KdV lift. (English) Zbl 1395.37046
The authors construct a Grassmann extension of a Yang-Baxter map which first appeared in the work of T. E. Kouloukas and V. G. Papageorgiou [J. Phys. A, Math. Theor. 42, No. 40, Article ID 404012, 12 p. (2009; Zbl 1217.37067)] 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 \(3 \times 3\) Lax matrix. Moreover, it is also shown 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, commutative analogs of the constructed Yang-Baxter map and its associated quad-graph system as well as their integrability is considered.
The paper is organised as follows. In Section II, they begin with some preliminaries on Grassmann algebras, Yang-Baxter maps, quadrilateral equations, and Lax representations.
After that, in Section III, the authors present the construction of the Grassmann extension of the lift of the dpKdV equation, and they show that it satisfies the Yang-Baxter equation.
In Section IV, it is shown that the constructed Yang-Baxter map can be reduced to a discrete quad-graph system, which constitutes a noncommutative extension of the dpKdV equation. The Lax representation of this system is derived from the corresponding Lax matrix of the Yang-Baxter map.
Section V deals with commutative analogs of the systems presented in Sections III and IV.
Finally, in Section VI, the authors close with some remarks and perspectives for future work.
The paper is organised as follows. In Section II, they begin with some preliminaries on Grassmann algebras, Yang-Baxter maps, quadrilateral equations, and Lax representations.
After that, in Section III, the authors present the construction of the Grassmann extension of the lift of the dpKdV equation, and they show that it satisfies the Yang-Baxter equation.
In Section IV, it is shown that the constructed Yang-Baxter map can be reduced to a discrete quad-graph system, which constitutes a noncommutative extension of the dpKdV equation. The Lax representation of this system is derived from the corresponding Lax matrix of the Yang-Baxter map.
Section V deals with commutative analogs of the systems presented in Sections III and IV.
Finally, in Section VI, the authors close with some remarks and perspectives for future work.
Reviewer: Eszter Gselmann (Debrecen)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 16T25 | Yang-Baxter equations |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 37K60 | Lattice dynamics; integrable lattice equations |
| 39A14 | Partial difference equations |
Citations:
Zbl 1217.37067References:
| [1] | Adler, V.; Bobenko, A.; Suris, Y., Classification of integrable equations on quad-graphs. The consistency approach, Commun. Math. Phys., 233, 513-543, (2003) · Zbl 1075.37022 · doi:10.1007/s00220-002-0762-8 |
| [2] | Adler, V.; Bobenko, A.; Suris, Y., Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings, Commun. Anal. Geom., 12, 967-1007, (2004) · Zbl 1065.14015 · doi:10.4310/cag.2004.v12.n5.a1 |
| [3] | Berezin, F., Intoduction to Superanalysis, (1987), D. Reidel: D. Reidel, Dordrecht |
| [4] | Bobenko, A.; Suris, Y., Integrable systems on quad-graphs, Int. Math. Res. Not., 11, 573-611, (2002) · Zbl 1004.37053 · doi:10.1155/s1073792802110075 |
| [5] | Bridgman, T.; Hereman, W.; Quispel, G. R. W.; 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) · Zbl 1306.37067 · doi:10.1007/s10208-012-9133-9 |
| [6] | Buchstaber, V., The Yang-Baxter transformation, Russ. Math. Surv., 53, 1343-1345, (1998) · Zbl 1015.17012 · doi:10.1070/rm1998v053n06abeh000094 |
| [7] | Doliwa, A., Non-commutative rational Yang-Baxter maps, Lett. Math. Phys., 104, 299-309, (2014) · Zbl 1306.37073 · doi:10.1007/s11005-013-0669-7 |
| [8] | Drinfeld, V., On some unsolved problems in quantum group theory, Lect. Notes Math., 1510, 1-8, (1992) · Zbl 0765.17014 · doi:10.1007/bfb0101175 |
| [9] | Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 169-209, (1999) · Zbl 0969.81030 · doi:10.1215/s0012-7094-99-10007-x |
| [10] | Grahovski, G.; Konstantinou-Rizos, S.; Mikhailov, A., Grassmann extensions of Yang-Baxter maps, J. Phys. A: Math. Theor., 49, 145202, (2016) · Zbl 1345.81055 · doi:10.1088/1751-8113/49/14/145202 |
| [11] | Grahovski, G.; Mikhailov, A., Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras, Phys. Lett. A, 377, 3254-3259, (2013) · Zbl 1293.35297 · doi:10.1016/j.physleta.2013.10.018 |
| [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) · Zbl 1334.39013 · doi:10.1143/jpsj.43.1424 |
| [13] | Kassotakis, P.; Nieszporski, M., On non-multiaffine consistent-around-the-cube lattice equations, Phys. Lett. A, 376, 3135-3140, (2012) · Zbl 1266.37038 · doi:10.1016/j.physleta.2012.10.009 |
| [14] | Konstantinou-Rizos, S.; Mikhailov, A., Darboux transformations, finite reduction groups and related Yang-Baxter maps, J. Phys. A: Math. Theor., 46, 425201, (2013) · Zbl 1276.81069 · doi:10.1088/1751-8113/46/42/425201 |
| [15] | Konstantinou-Rizos, S.; Mikhailov, A., Anticommutative extension of the Adler map, J. Phys. A: Math. Theor., 49, 30LT03, (2016) · Zbl 1353.16036 · doi:10.1088/1751-8113/49/30/30lt03 |
| [16] | Kouloukas, T. E.; Papageorgiou, V. G., Yang-Baxter maps with first-degree-polynomial 2 × 2 Lax matrices, J. Phys. A: Math. Theor., 42, 404012, (2009) · Zbl 1217.37067 · doi:10.1088/1751-8113/42/40/404012 |
| [17] | Kouloukas, T. E.; Papageorgiou, V. G., Entwining Yang-Baxter maps and integrable lattices, Banach Center Publ., 93, 163-175, (2011) · Zbl 1248.81087 · doi:10.4064/bc93-0-13 |
| [18] | Kouloukas, T. E.; Papageorgiou, V. G., Poisson Yang-Baxter maps with binomial Lax matrices, J. Math. Phys., 52, 073502, (2011) · Zbl 1317.37091 · doi:10.1063/1.3601520 |
| [19] | Kouloukas, T. E.; Tran, D., Poisson structures for lifts and periodic reductions of integrable lattice equations, J. Phys. A: Math. Theor., 48, 075202, (2015) · Zbl 1310.37028 · doi:10.1088/1751-8113/48/7/075202 |
| [20] | Mikhailov, A.; Papamikos, G.; Wang, J. P., Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere, Lett. Math. Phys., 106, 973-996, (2016) · Zbl 1358.35149 · doi:10.1007/s11005-016-0855-5 |
| [21] | Nijhoff, F.; Papageorgiou, V. G.; Capel, H. W.; Quispel, G. R. W., The lattice Gelfand-Dikii hierarchy, Inv. Probl., 8, 597-621, (1992) · Zbl 0763.35083 · doi:10.1088/0266-5611/8/4/010 |
| [22] | Nijhoff, F.; Capel, H., The discrete Korteweg-de Vries equation, Acta Appl. Math., 39, 133-158, (1995) · Zbl 0841.58034 · doi:10.1007/bf00994631 |
| [23] | Nijhoff, F.; Walker, A., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasgow Math. J., 43, 109-123, (2001) · Zbl 0990.39015 · doi:10.1017/s0017089501000106 |
| [24] | Papageorgiou, V. G.; Nijhoff, F. W.; Capel, H. W., Integrable mappings and nonlinear integrable lattice equations, Phys. Lett. A, 147, 106-114, (1990) · doi:10.1016/0375-9601(90)90876-p |
| [25] | Papageorgiou, V. G.; Tongas, A. G., Yang-Baxter maps and multi-field integrable lattice equations, J. Phys. A: Math. Theor., 40, 12677, (2007) · Zbl 1155.35466 · doi:10.1088/1751-8113/40/42/s12 |
| [26] | Papageorgiou, V. G. and Tongas, A. G., “Yang-Baxter maps associated to elliptic curves,” e-print (2009). |
| [27] | Papageorgiou, V. G.; Tongas, A. G.; Veselov, A. P., Yang-Baxter maps and symmetries of integrable equations on quad-graphs, J. Math. Phys., 47, 083502, (2006) · Zbl 1112.82017 · doi:10.1063/1.2227641 |
| [28] | Sklyanin, E., Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl., 16, 263-270, (1982) · Zbl 0513.58028 · doi:10.1007/bf01077848 |
| [29] | Sklyanin, E., Classical limits of SU(2)-invariant solutions of the Yang-Baxter equation, J. Sov. Math., 40, 93-107, (1988) · Zbl 0636.58041 · doi:10.1007/bf01084941 |
| [30] | Sklyanin, E., Separation of variables, new trends, Prog. Theor. Phys. Suppl., 118, 35-60, (1995) · Zbl 0868.35002 · doi:10.1143/ptps.118.35 |
| [31] | Suris, Y.; Veselov, A., Lax matrices for Yang-Baxter maps, J. Nonlinear Math. Phys., 10, 223-230, (2003) · Zbl 1362.39016 · doi:10.2991/jnmp.2003.10.s2.18 |
| [32] | Veselov, A., Yang-Baxter maps and integrable dynamics, Phys. Lett. A, 314, 214-221, (2003) · Zbl 1051.81014 · doi:10.1016/s0375-9601(03)00915-0 |
| [33] | Veselov, A., Yang-Baxter maps: Dynamical point of view, Math. Soc. Jpn. Mem., 17, 145-167, (2007) · Zbl 1232.81023 · doi:10.2969/msjmemoirs/01701C060 |
| [34] | Xue, L. L.; Levi, D.; Liu, Q. P., Supersymmetric KdV equation: Darboux transformation and discrete systems, J. Phys. A: Math. Theor., 46, 502001, (2013) · Zbl 1286.35224 · doi:10.1088/1751-8113/46/50/502001 |
| [35] | Xue, L. L.; Liu, Q. P., Bäcklund-Darboux transformations and discretizations of super KdV equation, SIGMA, 10, 045, (2014) · Zbl 1291.35319 · doi:10.3842/SIGMA.2014.045 |
| [36] | Xue, L. L.; Liu, Q. P., A supersymmetric AKNS problem and its Darboux-Bäcklund transformations and discrete systems, Stud. Appl. Math., 135, 35-62, (2015) · Zbl 1328.35205 · doi:10.1111/sapm.12080 |
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