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

SIGMA 16 (2020), 060, 30 pages      arXiv:1912.00713      https://doi.org/10.3842/SIGMA.2020.060

### Multi-Component Extension of CAC Systems

Dan-Da Zhang a, Peter H. van der Kamp b and Da-Jun Zhang c
a) School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China
b) Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
c) Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China

Received December 07, 2019, in final form June 14, 2020; Published online July 01, 2020

Abstract
In this paper an approach to generate multi-dimensionally consistent $N$-component systems is proposed. The approach starts from scalar multi-dimensionally consistent quadrilateral systems and makes use of the cyclic group. The obtained $N$-component systems inherit integrable features such as Bäcklund transformations and Lax pairs, and exhibit interesting aspects, such as nonlocal reductions. Higher order single component lattice equations (on larger stencils) and multi-component discrete Painlevé equations can also be derived in the context, and the approach extends to $N$-component generalizations of higher dimensional lattice equations.

Key words: lattice equations; consistency around the cube; cyclic group; multi-component; Lax pair; Bäcklund transformation; nonlocal.

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References

1. Ablowitz M.J., Musslimani Z.H., Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett. 110 (2013), 064105, 5 pages.
2. Adler V.E., On the structure of the Bäcklund transformations for the relativistic lattices, J. Nonlinear Math. Phys. 7 (2000), 34-56, arXiv:nlin.SI/0001072.
3. Adler V.E., Discrete equations on planar graphs, J. Phys. A: Math. Gen. 34 (2001), 10453-10460.
4. 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.
5. 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.
6. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable discrete equations of octahedron type, Int. Math. Res. Not. 2012 (2012), 1822-1889, arXiv:1011.3527.
7. Atkinson J., Bäcklund transformations for integrable lattice equations, J. Phys. A: Math. Theor. 41 (2008), 135202, 8 pages, arXiv:0801.1998.
8. Atkinson J., Integrable lattice equations: connection to the Möbius group, Bäcklund transformations and solutions, Ph.D. Thesis, University of Leeds, 2008, available at http: etheses.whiterose.ac.uk/9081.
9. Atkinson J., Nieszporski M., Multi-quadratic quad equations: integrable cases from a factorized-discriminant hypothesis, Int. Math. Res. Not. 2014 (2014), 4215-4240, arXiv:1204.0638.
10. Babalic C.N., Carstea A.S., Coupled Ablowitz-Ladik equations with branched dispersion, J. Phys. A: Math. Theor. 50 (2017), 415201, 13 pages, arXiv:1705.10975.
11. Bobenko A.I., Suris Yu.B., Integrable noncommutative equations on quad-graphs. The consistency approach, Lett. Math. Phys. 61 (2002), 241-254, arXiv:nlin.SI/0206010.
12. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), 573-611, arXiv:nlin.SI/0110004.
13. Boll R., Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 18 (2011), 337-365, arXiv:1009.4007.
14. 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 (2013), 517-544, arXiv:1308.5473.
15. Carstea A.S., Tokihiro T., Coupled discrete KdV equations and modular genetic networks, J. Phys. A: Math. Theor. 48 (2015), 055205, 12 pages.
16. Chen K., Zhang C., Zhang D.-J., Squared eigenfunction symmetry of the D$\Delta$mKP hierarchy and its constraint, arXiv:1904.08108.
17. Doliwa A., Grinevich P., Nieszporski M., Santini P.M., Integrable lattices and their sublattices: from the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme, J. Math. Phys. 48 (2007), 013513, 28 pages, arXiv:nlin.SI/0410046.
18. Duffin R.J., Basic properties of discrete analytic functions, Duke Math. J. 23 (1956), 335-363.
19. Field C.M., Nijhoff F.W., Capel H.W., Exact solutions of quantum mappings from the lattice KdV as multi-dimensional operator difference equations, J. Phys. A: Math. Gen. 38 (2005), 9503-9527.
20. Fordy A.P., Xenitidis P., ${\mathbb Z}_N$ graded discrete Lax pairs and integrable difference equations, J. Phys. A: Math. Theor. 50 (2017), 165205, 30 pages, arXiv:1411.6059.
21. Fu W., Nijhoff F.W., Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 473 (2017), 20160195, 22 pages, arXiv:1612.04711,.
22. Fu W., Zhang D.-J., Zhou R.-G., A class of two-component Adler-Bobenko-Suris lattice equations, Chinese Phys. Lett. 31 (2004), 090202, 5 pages.
23. Gubbiotti G., Scimiterna C., Yamilov R.I., Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations II: General solutions, SIGMA 14 (2018), 008, 51 pages, arXiv:1704.05805.
24. Hietarinta J., Boussinesq-like multi-component lattice equations and multi-dimensional consistency, J. Phys. A: Math. Theor. 44 (2011), 165204, 22 pages, arXiv:1011.1978.
25. Hietarinta J., Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax, J. Nonlinear Math. Phys. 26 (2019), 358-389, arXiv:1806.08511.
26. Hietarinta J., Zhang D.-J., On the relations of the dsG and lpmKdV, unpublished.
27. Hietarinta J., Zhang D.-J., Soliton solutions for ABS lattice equations. II. Casoratians and bilinearization, J. Phys. A: Math. Theor. 42 (2009), 404006, 30 pages, arXiv:0903.1717.
28. Hietarinta J., Zhang D.-J., Multisoliton solutions to the lattice Boussinesq equation, J. Math. Phys. 51 (2010), 033505, 12 pages, arXiv:0906.3955.
29. Hietarinta J., Zhang D.-J., Soliton taxonomy for a modification of the lattice Boussinesq equation, SIGMA 7 (2011), 061, 14 pages, arXiv:1105.4413.
30. Hirota R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981), 3785-3791.
31. Joshi N., Lobb S., Nolan M., Constructing initial value spaces of lattice equations, arXiv:1807.06162.
32. Joshi N., Nakazono N., Shi Y., Reflection groups and discrete integrable systems, J. Integrable Systems 1 (2016), xyw006, 37 pages, arXiv:1605.01171.
33. 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.
34. Kassotakis P., Nieszporski M., Papageorgiou V., Tongas A., Tetrahedron maps and symmetries of three dimensional integrable discrete equations, J. Math. Phys. 60 (2019), 123503, 18 pages, arXiv:1908.03019.
35. Kassotakis P., Nieszporski M., Papageorgiou V., Tongas A., Integrable two-component systems of difference equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 476 (2020), 20190668, 22 pages, arXiv:1908.02413.
36. Kels A.P., Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations, J. Stat. Phys. 176 (2019), 1375-1408, arXiv:1812.10893.
37. King A.D., Schief W.K., Bianchi hypercubes and a geometric unification of the Hirota and Miwa equations, Int. Math. Res. Not. 2015 (2015), 6842-6878.
38. Levi D., Martina L., Winternitz P., Structure preserving discretizations of the Liouville equation and their numerical tests, SIGMA 11 (2015), 080, 20 pages, arXiv:1504.01953.
39. Levi D., Ragnisco O., Bruschi M., Continuous and discrete matrix Burgers' hierarchies, Nuovo Cimento B 74 (1983), 33-51.
40. Mercat C., Discrete Riemann surfaces and the Ising model, Comm. Math. Phys. 218 (2001), 177-216, arXiv:0909.3600.
41. Miwa T., On Hirota's difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 9-12.
42. Nakazono N., Reduction of lattice equations to the Painlevé equations: $\rm P_{IV}$ and $\rm P_V$, J. Math. Phys. 59 (2018), 022702, 18 pages, arXiv:1703.09215.
43. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, arXiv:nlin.SI/0110027.
44. Nijhoff F.W., Atkinson J., Hietarinta J., Soliton solutions for ABS lattice equations. I. Cauchy matrix approach, J. Phys. A: Math. Theor. 42 (2009), 404005, 34 pages, arXiv:0902.4873.
45. Nijhoff F.W., Capel H.W., The direct linearisation approach to hierarchies of integrable PDEs in $2+1$ dimensions. I. Lattice equations and the differential-difference hierarchies, Inverse Problems 6 (1990), 567-590.
46. Nijhoff F.W., Papageorgiou V.G., Capel H.W., Integrable time-discrete systems: lattices and mappings, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 312-325.
47. Nijhoff F.W., Papageorgiou V.G., Capel H.W., Quispel G.R.W., The lattice Gel'fand-Dikii hierarchy, Inverse Problems 8 (1992), 597-621.
48. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J. 43 (2001), 109-123, arXiv:nlin.SI/0001054.
49. Nimmo J.J.C., Darboux transformations and the discrete KP equation, J. Phys. A: Math. Gen. 30 (1997), 8693-8704.
50. Papageorgiou V., Tongas A., Yang-Baxter maps associated to elliptic curves, arXiv:0906.3258.
51. Suris Yu.B., A discrete-time relativistic Toda lattice, J. Phys. A: Math. Gen. 29 (1996), 451-465, arXiv:solv-int/9510007.
52. Tongas A., Nijhoff F., The Boussinesq integrable system: compatible lattice and continuum structures, Glasg. Math. J. 47 (2005), 205-219, arXiv:nlin.SI/0402053.
53. Tsarev S.P., Wolf T., Classification of three-dimensional integrable scalar discrete equations, Lett. Math. Phys. 84 (2008), 31-39, arXiv:0706.2464.
54. van der Kamp P.H., Quispel G.R.W., Zhang D.-J., Duality for discrete integrable systems II, J. Phys. A: Math. Theor. 51 (2018), 365202, 13 pages, arXiv:1711.05886.
55. van der Kamp P.H., Zhang D.-J., Quispel G.R.W., On the relation between the dual AKP equation and an equation by King and Schief, and its $N$-soliton solution, arXiv:1912.02299.
56. Viallet C.M., Integrable lattice maps: $Q_V$, a rational version of $Q_4$, Glasg. Math. J. 51 (2009), 157-163, arXiv:0802.0294.
57. Walker A.J., Similarity reductions and integrable lattice equations, Ph.D. Thesis, University of Leeds, 2001.
58. Xenitidis P.D., Papageorgiou V.G., Symmetries and integrability of discrete equations defined on a black-white lattice, J. Phys. A: Math. Theor. 42 (2009), 454025, 13 pages, arXiv:0903.3152.
59. Zhang D., Zhang D.-J., Rational solutions to the ABS list: transformation approach, SIGMA 13 (2017), 078, 24 pages, arXiv:1702.01266.
60. Zhang D.-J., The Sylvester equation, Cauchy matrices and matrix discrete systems, https: www.newton.ac.uk/files/seminar/20130708140014301-153640.pdf.
61. Zhao S.-L., Zhang D.-J., Rational solutions to ${\rm Q}3_{\delta}$ in the Adler-Bobenko-Suris list and degenerations, J. Nonlinear Math. Phys. 26 (2019), 107-132.