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

SIGMA 13 (2017), 057, 17 pages      arXiv:1705.01094
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

On Reductions of the Hirota-Miwa Equation

Andrew N.W. Hone, Theodoros E. Kouloukas and Chloe Ward
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK

Received May 02, 2017, in final form July 17, 2017; Published online July 23, 2017

The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or half-integers, which produce bilinear ordinary difference equations of Somos/Gale-Robinson type. Here it is explained how to obtain Lax pairs and presymplectic structures for these reductions, in order to demonstrate Liouville integrability of some associated maps, certain of which are related to reductions of discrete Toda and discrete KdV equations.

Key words: Hirota-Miwa equation; Liouville integrable maps; Somos sequences; cluster algebras.

pdf (424 kb)   tex (25 kb)


  1. Braden H.W., Enolskii V.Z., Hone A.N.W., Bilinear recurrences and addition formulae for hyperelliptic sigma functions, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 46-62, math.NT/0501162.
  2. Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. III, J. Phys. Soc. Japan 52 (1983), 388-393.
  3. Doliwa A., Lin R., Discrete KP equation with self-consistent sources, Phys. Lett. A 378 (2014), 1925-1931, arXiv:1310.4636.
  4. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930, math.AG/0311245.
  5. Fomin S., Zelevinsky A., The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), 119-144, math.CO/0104241.
  6. Fordy A.P., Hone A.N.W., Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys. 325 (2014), 527-584, arXiv:1207.6072.
  7. Fordy A.P., Marsh R.J., Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin. 34 (2011), 19-66, arXiv:0904.0200.
  8. Gekhtman M., Shapiro M., Vainshtein A., Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), 291-311, math.QA/0309138.
  9. Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2016.
  10. Hirota R., Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation, J. Phys. Soc. Japan 43 (1977), 1424-1433.
  11. Hirota R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981), 3785-3791.
  12. Hone A.N.W., Inoue R., Discrete Painlevé equations from Y-systems, J. Phys. A: Math. Theor. 47 (2014), 474007, 26 pages, arXiv:1405.5379.
  13. Hone A.N.W., Kouloukas T.E., Quispel G.R.W., Some integrable maps and their Hirota bilinear forms, in preparation.
  14. Hone A.N.W., van der Kamp P.H., Quispel G.R.W., Tran D.T., Integrability of reductions of the discrete Korteweg-de Vries and potential Korteweg-de Vries equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), 20120747, 23 pages, arXiv:1211.6958.
  15. Inoue R., Nakanishi T., Difference equations and cluster algebras I: Poisson bracket for integrable difference equations, in Infinite Analysis 2010 - Developments in Quantum Integrable Systems, RIMS Kôkyûroku Bessatsu, Vol. B28, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, 63-88, arXiv:1012.5574.
  16. Jeong I.-J., Musiker G., Zhang S., Gale-Robinson sequences and brane tilings, in 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Discrete Math. Theor. Comput. Sci. Proc., AS, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2013, 707-718.
  17. Krichever I., Lipan O., Wiegmann P., Zabrodin A., Quantum integrable models and discrete classical Hirota equations, Comm. Math. Phys. 188 (1997), 267-304.
  18. Maeda S., Completely integrable symplectic mapping, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 198-200.
  19. Maruno K., Quispel G.R.W., Construction of integrals of higher-order mappings, J. Phys. Soc. Japan 75 (2006), 123001, 5 pages, nlin.SI/0611020.
  20. Mase T., Investigation into the role of the Laurent property in integrability, J. Math. Phys. 57 (2016), 022703, 21 pages, arXiv:1505.01722.
  21. Miwa T., On Hirota's difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 9-12.
  22. Mumford D., Tata lectures on theta. II. Jacobian theta functions and differential equations, Progress in Mathematics, Vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984.
  23. Taimanov I.A., Secants of abelian varieties, theta functions and soliton equations, Russian Math. Surveys 52 (1997), 147-218, alg-geom/9609019.
  24. Tran D.T., van der Kamp P.H., Quispel G.R.W., Poisson brackets of mappings obtained as $(q,-p)$ reductions of lattice equations, Regul. Chaotic Dyn. 21 (2016), 682-696, arXiv:1608.08010.
  25. van der Kamp P.H., Quispel G.R.W., The staircase method: integrals for periodic reductions of integrable lattice equations, J. Phys. A: Math. Theor. 43 (2010), 465207, 34 pages, arXiv:1005.2071.
  26. Vekslerchik V.E., Finite-genus solutions for the Hirota's bilinear difference equation, nlin.SI/0002005.
  27. Veselov A.P., Integrable mappings, Russian Math. Surveys 46 (1991), no. 5, 1-51.
  28. Ward C., Discrete integrability and nonlinear recurrences with the Laurent property, Ph.D. Thesis, University of Kent, 2013.
  29. Zabrodin A.V., Hirota difference equations, Theoret. and Math. Phys. 113 (1997), 1347-1392, solv-int/9704001.

Previous article  Next article   Contents of Volume 13 (2017)