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A Revisit to the ABS $\mathrm{H2}$ Equation
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we revisit the Adler–Bobenko–Suris $\mathrm{H2}$ equation. The $\mathrm{H2}$ 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.
Keywords: consistent around cube, Cauchy matrix approach, continuum limit, KdV equation.
Mots-clés : $\mathrm{H2}$ equation 
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     author = {Aye Aye Cho and Maebel Mesfun and Da-Jun Zhang},
     title = {A {Revisit} to the {ABS} $\mathrm{H2}$ {Equation}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a92/}
}
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Aye Aye Cho; Maebel Mesfun; Da-Jun Zhang. A Revisit to the ABS $\mathrm{H2}$ Equation. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a92/

[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 | DOI

[2] Atkinson J., “Bäcklund transformations for integrable lattice equations”, J. Phys. A: Math. Theor., 41 (2008), 135202, 8 pp., arXiv: 0801.1998 | DOI

[3] Bobenko A. I., Suris Yu.B., “Integrable systems on quad-graphs”, Int. Math. Res. Not., 2002 (2002), 573–611, arXiv: nlin.SI/0110004 | DOI

[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 | DOI

[5] Chen Z.-Y., Bi J.-B., Chen D.-Y., “Novel solutions of KdV equation”, Commun. Theor. Phys. (Beijing), 41 (2004), 397–399 | DOI

[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 | DOI

[7] Hietarinta J., Joshi N., Nijhoff F. W., Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2016 | DOI

[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 pp., arXiv: 0903.1717 | DOI

[9] Hirota R., “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27 (1971), 1192–1194 | DOI

[10] Hirota R., “A new form of Bäcklund transformations and its relation to the inverse scattering problem”, Progr. Theoret. Phys., 52 (1974), 1498–1512 | DOI

[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 pp., arXiv: 1908.02413 | DOI

[12] Levi D., Benguria R., “Bäcklund transformations and nonlinear differential difference equations”, Proc. Nat. Acad. Sci. USA, 77 (1980), 5025–5027 | DOI

[13] Nijhoff F. W., “Lax pair for the Adler (lattice Krichever–Novikov) system”, Phys. Lett. A, 297 (2002), 49–58, arXiv: nlin.SI/0110027 | DOI

[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 pp., arXiv: 0902.4873 | DOI

[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 | DOI

[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 | DOI

[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 pp., arXiv: 1811.01855 | DOI

[19] Wiersma G. L., Capel H. W., “Lattice equations, hierarchies and Hamiltonian structures”, Phys. A, 142 (1987), 199–244 | DOI

[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 | DOI

[21] Zhang D., Zhang D.-J., “Rational solutions to the ABS list: transformation approach”, SIGMA, 13 (2017), 078, 24 pp., arXiv: 1702.01266 | DOI

[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 | DOI