×

Cauchy problem for integrable discrete equations on quad-graphs. (English) Zbl 1067.37086

Summary: Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well posed. In the basic example of the discrete Korteweg-de Vries equation, an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice is discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K60 Lattice dynamics; integrable lattice equations
52C23 Quasicrystals and aperiodic tilings in discrete geometry
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Adler, V. E.: Discrete equations on planar graphs, J. Phys. A: Math. Gen. 34 (2001), 10453–10460. · Zbl 1001.37081 · doi:10.1088/0305-4470/34/48/310
[2] Adler, V. E., Bobenko, A. I. and Suris, Yu. B.: Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233(3) (2003), 513–543. · Zbl 1075.37022
[3] Bianchi, L.: Vorlesungen über Differenzialgeometrie, Teubner, Leipzig, 1899.
[4] Bobenko, A. I. and Suris, Yu. B.: Integrable systems on quad-graphs, Int. Math. Res. Notices 11 (2002), 573–611. · Zbl 1004.37053 · doi:10.1155/S1073792802110075
[5] Bobenko, A. I., Hoffmann, T. and Suris, Yu. B.: Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and Lax equations on the regular triangular lattice, Int. Math. Res. Notices (3) (2002), 111–164. · Zbl 0993.52004
[6] Capel, H. W., Nijhoff, F. W. and Papageorgiou, V. G.: Complete integrability of Lagrangian mappings and lattices of KdV type, Phys. Lett. A 155 (1991), 377–387. · doi:10.1016/0375-9601(91)91043-D
[7] Dolbilin, N. P., Sedrakyan, A. G., Shtan’ko, M. A. and Shtogrin, M. I.: Topology of a family of parametrizations of two-dimensional cycles arising in the three-dimensional Ising model, Dokl. Akad. Nauk SSSR 295(1) (1987), 19–23 [English translation: Soviet Math. Dokl. 36(1) (1988), 11–15]. · Zbl 0645.55003
[8] Drinfeld, V. G.: On some unsolved problems in quantum group theory, In: Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer, 1992, pp. 1–8.
[9] Dynnikov, I. A. and Novikov, S. P.: Laplace transformations and simplicial connections, Uspekhi Mat. Nauk 52(6) (1997), 157–158 [English translation: Russian Math. Surveys 52(6) (1997), 1294–1295]. · Zbl 0928.39009
[10] Dynnikov, I. A. and Novikov, S. P.: Geometry of the triangle equation on two-manifolds, math-ph/0208041.
[11] Hirota, R.: Nonlinear partial difference equations. I. A difference analog of the Korteweg–de Vries equation. III. Discrete sine-Gordon equation, J. Phys. Soc. Japan 43 (1977), 1423–1433, 2079–2086. · Zbl 1334.39013
[12] King, A. and Schief, W.: Tetrahedra, octahedra and cubo-octahedra: Integrable geometry of multiratios, J. Phys. A: Math. Gen. 36 (2003), 785–802. · Zbl 1057.51022 · doi:10.1088/0305-4470/36/3/313
[13] Konopelchenko, B. G. and Schief, W. K.: Three-dimensional integrable lattices in Euclidean spaces: Conjugacy and orthogonality, Proc. Roy. Soc. A 454 (1998), 3075–3104. · Zbl 1050.37034 · doi:10.1098/rspa.1998.0292
[14] Konopelchenko, B. G. and Schief, W. K.: Menelaus’ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy, J. Phys. A: Math. Gen. 35 (2002), 6125–6144. · Zbl 1039.37052 · doi:10.1088/0305-4470/35/29/313
[15] Korepin, V. E.: Exactly solvable spin models for quasicrystals, JETP 92(3) (1987), 1082–1089.
[16] Korepin, V. E.: Completely integrable models in quasicrystals, Comm. Math. Phys. 110(1) (1987), 157–171. · Zbl 0614.52009 · doi:10.1007/BF01209021
[17] Krichever, I. M. and Novikov, S. P.: Trivalent graphs and solitons, Uspekhi Mat. Nauk 54(1) (1999), 149–150 [English translation: Russian Math. Surveys 54(1) (1999), 1248–1249]. · Zbl 0981.37023
[18] Miwa, T.: On Hirota’s difference equations, Proc. Japan Acad., Ser. A: Math. Sci. 58(1) (1982), 9–12. · Zbl 0508.39009 · doi:10.3792/pjaa.58.9
[19] Nijhoff, F. W.: Lax pair for the Adler (lattice Krichever–Novikov) system, Phys. Lett. A 297(1–2) (2002), 49–58. · Zbl 0994.35105 · doi:10.1016/S0375-9601(02)00287-6
[20] Nijhoff, F. W. and Capel, H. W.: The discrete Korteweg–de Vries equation, Acta Appl. Math. 39 (1995), 133–158. · Zbl 0841.58034 · doi:10.1007/BF00994631
[21] Nijhoff, F. W. and Walker, A. J.: The discrete and continuous Painlevé hierarchy and the Garnier system, Glasgow Math. J. 43A (2001), 109–123. · Zbl 0990.39015 · doi:10.1017/S0017089501000106
[22] Novikov, S. P. and Dynnikov, I. A.: Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-manifolds, Uspekhi Mat. Nauk 52(5) (1997), 175–234 [English translation: Russian Math. Surveys 52(5) (1997), 1057–1116]. · Zbl 0928.35107
[23] Novikov, S. P.: The Schrödinger operators on graphs and topology, Uspekhi Mat. Nauk 52(6) (1997), 177–178 [English translation: Russian Math. Surveys 52(6) (1997), 1320–1321]. · Zbl 0925.34114
[24] Papageorgiou, V. G., Nijhoff, F. W. and Capel, H. W.: Integrable mappings and nonlinear integrable lattice equations, Phys. Lett. A 147(2–3) (1990), 106–114. · Zbl 0717.35079 · doi:10.1016/0375-9601(90)90876-P
[25] Quispel, G. R. W., Nijhoff, F. W., Capel, H. W. and van der Linden, J.: Linear integral equations and nonlinear difference-difference equations, Physica A 125 (1984), 344–380. · Zbl 0598.45009 · doi:10.1016/0378-4371(84)90059-1
[26] Senechal, M.: Quasicrystals and Geometry, Cambridge University Press, Cambridge, 1995. · Zbl 0828.52007
[27] Veselov, A. P.: Yang–Baxter maps and integrable dynamics, Phys. Lett. A 314 (2003), 214–221. · Zbl 1051.81014 · doi:10.1016/S0375-9601(03)00915-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.