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

SIGMA 17 (2021), 093, 19 pages      arXiv:2106.12835

A Revisit to the ABS H2 Equation

Aye Aye Cho, Maebel Mesfun and Da-Jun Zhang
Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China

Received June 25, 2021, in final form October 13, 2021; Published online October 18, 2021

In this paper we revisit the Adler-Bobenko-Suris H2 equation. The H2 equation is linearly related to the $S^{(0,0)}$ and $S^{(1,0)}$ variables in the Cauchy matrix scheme. We elaborate the coupled quad-system of $S^{(0,0)}$ and $S^{(1,0)}$ in terms of their 3-dimensional consistency, Lax pair, bilinear form and continuum limits. It is shown that $S^{(1,0)}$ itself satisfies a 9-point lattice equation and in continuum limit $S^{(1,0)}$ is related to the eigenfunction in the Lax pair of the Korteweg-de Vries equation.

Key words: H2 equation; consistent around cube; Cauchy matrix approach; continuum limit; KdV equation.

pdf (483 kb)   tex (24 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. Atkinson J., Bäcklund transformations for integrable lattice equations, J. Phys. A: Math. Theor. 41 (2008), 135202, 8 pages, arXiv:0801.1998.
  3. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), 573-611, arXiv:nlin.SI/0110004.
  4. 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.
  5. Chen Z.-Y., Bi J.-B., Chen D.-Y., Novel solutions of KdV equation, Commun. Theor. Phys. (Beijing) 41 (2004), 397-399.
  6. Freeman N.C., Nimmo J.J.C., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique, Phys. Lett. A 95 (1983), 1-3.
  7. Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2016.
  8. 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.
  9. Hirota R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971), 1192-1194.
  10. Hirota R., A new form of Bäcklund transformations and its relation to the inverse scattering problem, Progr. Theoret. Phys. 52 (1974), 1498-1512.
  11. Kassotakis P., Nieszporski M., Papageorgiou V., Tongas A., Integrable two-component systems of difference equations, Proc. Roy. Soc. Ser. A 476 (2020), 20190668, 22 pages, arXiv:1908.02413.
  12. Levi D., Benguria R., Bäcklund transformations and nonlinear differential difference equations, Proc. Nat. Acad. Sci. USA 77 (1980), 5025-5027.
  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., 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.
  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. Sylvester J.J., Sur l'equation en matrices $px=xq$, C.R. Acad. Sci. Paris 99 (1884), 67-71, 115-117.
  18. Vermeeren M., A variational perspective on continuum limits of ABS and lattice GD equations, SIGMA 15 (2019), 044, 35 pages, arXiv:1811.01855.
  19. Wiersma G.L., Capel H.W., Lattice equations, hierarchies and Hamiltonian structures, Phys. A 142 (1987), 199-244.
  20. Xu D.-D., Zhang D.-J., Zhao S.-L., The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation, J. Nonlinear Math. Phys. 21 (2014), 382-406, arXiv:1401.5949.
  21. Zhang D., Zhang D.-J., Rational solutions to the ABS list: transformation approach, SIGMA 13 (2017), 078, 24 pages, arXiv:1702.01266.
  22. Zhang D.-J., Zhao S.-L., Solutions to ABS lattice equations via generalized Cauchy matrix approach, Stud. Appl. Math. 131 (2013), 72-103, arXiv:1208.3752.

Previous article  Next article  Contents of Volume 17 (2021)