Connection between the symmetric discrete AKP system and bilinear ABS lattice equations. (English) Zbl 07864744
Summary: In this paper, we show that all the bilinear Adler-Bobenko-Suris (ABS) equations (except Q2 and Q4) can be obtained from symmetric discrete AKP system by taking proper reductions and continuum limits. Among the bilinear ABS equations, a simpler bilinear form of the ABS H2 equation is given. In addition, an 8-point 3-dimensional lattice equation and an 8-point 4-dimensional lattice equation are obtained as by-products. Both of them can be considered as extensions of the symmetric discrete AKP equation.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 39A14 | Partial difference equations |
Keywords:
Adler-Bobenko-Suris equations; symmetric discrete AKP system; bilinear equation; reduction; continuum limitReferences:
| [1] | Hietarinta, J.; Joshi, N.; Nijhoff, F. W., Discrete Systems and Integrablity, 2016, Camb. Univ. Press: Camb. Univ. Press Cambridge · Zbl 1362.37130 |
| [2] | Hirota, R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan, 50, 3785-3791, 1981 |
| [3] | Miwa, T., On Hirota’s difference equations, Proc. Japan Acad. Ser. A Math. Sci., 58, 9-12, 1982 · Zbl 0508.39009 |
| [4] | Date, E.; Jimbo, M.; Miwa, T., Method for generating discrete soliton equations. III, J. Phys. Soc. Japan, 52, 388-393, 1983 · Zbl 0571.35105 |
| [5] | Ramani, A.; Grammaticos, B.; Satsuma, J., Integrability of multidimensional discrete systems, Phys. Lett. A, 169, 323-328, 1992 |
| [6] | Zabrodin, A., Hirota’s difference equations, Theoret. Math. Phys., 113, 1317-1392, 1997 |
| [7] | Adler, V. E.; Bobenko, A. I.; Suris, Y. B., Classification of integrable discrete equations of octahedron type, Int. Math. Res. Not. IMRN, 2012, 1822-1889, 2012 · Zbl 1277.39006 |
| [8] | Ohta, Y.; Hirota, R.; Tsujimoto, S.; Imai, T., Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy, J. Phys. Soc. Japan, 62, 1872-1886, 1993 · Zbl 0972.37536 |
| [9] | Li, S. S.; Nijhoff, F. W.; Sun, Y. Y.; Zhang, D. J., Symmetric discrete AKP and BKP equations, J. Phys. A, 54, Article 075201 pp., 2021, (19pp) · Zbl 1519.39003 |
| [10] | Alexandrov, A., KdV solves BKP, Proc. Natl. Acad. Sci., 118, 1-2, 2021 |
| [11] | Adler, V. E.; Bobenko, A. I.; Suris, Yu. B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys., 233, 513-543, 2003 · Zbl 1075.37022 |
| [12] | Atkinson, J.; Hietarinta, J.; Nijhoff, F. W., Seed and soliton solutions for Adler’s lattice equation, J. Phys. A, 40, F1-F8, 2007 · Zbl 1106.37048 |
| [13] | Atkinson, J.; Hietarinta, J.; Nijhoff, F. W., Soliton solutions for Q3, J. Phys. A, 41, Article 142001 pp., 2008, (11pp) · Zbl 1136.37036 |
| [14] | Nijhoff, F. W.; Atkinson, J.; Hietarinta, J., Soliton solutions for ABS lattice equations: I: Cauchy matrix approach, J. Phys. A, 42, Article 404005 pp., 2009, (34pp) · Zbl 1184.35281 |
| [15] | Hietarinta, J.; Zhang, D. J., Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization, J. Phys. A, 42, Article 404006 pp., 2009, (30pp) · Zbl 1184.35276 |
| [16] | Nijhoff, F. W.; Atkinson, J., Elliptic N-soliton solutions of ABS lattice equations, Int. Math. Res. Not. IMRN, 2010, 3837-3895, 2010 · Zbl 1217.37068 |
| [17] | Atkinson, J.; Nijhoff, F. W., A constructive approach to the soliton solutions of integrable quadrilateral lattice equations, Comm. Math. Phys., 299, 283-304, 2010 · Zbl 1198.35205 |
| [18] | Butler, S.; Joshi, N., An inverse scattering transform for the lattice potential KdV equation, Inverse Prob., 26, Article 115012 pp., 2010, (28pp) · Zbl 1204.35144 |
| [19] | Butler, S., Multidimensional inverse scattering of integrable lattice equations, Nonlinearity, 25, 1613-1634, 2012 · Zbl 1251.39003 |
| [20] | Zhang, D. J.; Zhao, S. L., Solutions to the ABS lattice equations via generalized Cauchy matrix approach, Stud. Appl. Math., 131, 72-103, 2013 · Zbl 1338.37113 |
| [21] | Shi, Y.; Nimmo, J. J.C.; Zhang, D. J., Darboux and binary Darboux transformations for discrete integrable systems I. Discrete potential KdV equation, J. Phys. A, 47, Article 025205 pp., 2014, (11pp) · Zbl 1285.35098 |
| [22] | Zhang, D. D.; Zhang, D. J., Rational solutions to the ABS list: Transformation approach, SIGMA, 13, 078, 2017, (24pp) · Zbl 1383.35185 |
| [23] | Zhao, S. L.; Zhang, D. J., Rational solutions to Q \(3_\delta\) in the Adler-Bobenko-Suris list and degenerations, J. Nonlinear Math. Phys., 26, 107-132, 2019 · Zbl 1417.37242 |
| [24] | Vermeeren, M., A variational perspective on continuum limits of ABS and lattice GD equations, SIGMA, 15, 044, 2019, (35pp) · Zbl 1432.37097 |
| [25] | Chou, A. A.; Mesfun, M.; Zhang, D. J., A revisit to the ABS H2 equation, SIGMA, 17, 093, 2021, (19pp) · Zbl 1477.35216 |
| [26] | Atkinson, J., Bäcklund transformations for integrable lattice equations, J. Phys. A, 41, Article 135202 pp., 2008, (8pp) · Zbl 1148.82006 |
| [27] | Hietarinta, J., Searching for CAC-maps, J. Nonlinear Math. Phys., 12, 223-230, 2005 · Zbl 1091.37020 |
| [28] | Nimmo, J. J.C.; Freeman, N. C., Rational solutions of the Korteweg-de Vries equation in Wronskian form, Phys. Lett. A, 96, 443-446, 1983 |
| [29] | Hirota, R., A new form of Bäcklund transformations and its relation to the inverse scattering problem, Prog. Theor. Phys., 52, 1498-1512, 1974 · Zbl 1168.37322 |
| [30] | Nijhoff, F. W.; Quispel, G. R.W.; Capel, H. W., Direct linearization of nonlinear difference-difference equations, Phys. Lett. A, 97, 125-128, 1983 |
| [31] | Date, E.; Jimbo, M.; Miwa, T., Method for generating discrete soliton equations. II, J. Phys. Soc. Japan, 51, 4125-4131, 1982 |
| [32] | Chen, Z. Y.; Bi, J. B.; Chen, D. Y., Novel solutions of KdV equation, Commun. Theor. Phys., 41, 397-399, 2004 · Zbl 1167.35462 |
| [33] | Hietarinta, J.; Zhang, D. J., Discrete Boussinesq-type equations, (Euler, N.; Zhang, D. J., Nonlinear Systems and their Remarkable Mathematical Structures, Vol. 3, 2021, CRC Press, Taylor & Francis: CRC Press, Taylor & Francis Boca Raton), 54-101 |
| [34] | Nijhoff, F. W.; Sun, Y. Y.; Zhang, D. J., Elliptic solutions of Boussinesq type lattice equations and the elliptic \(N\) th root of unity, Comm. Math. Phys., 399, 599-650, 2023 · Zbl 1518.39003 |
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