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Acta Applicandae Mathematica

, Volume 84, Issue 2, pp 237–262 | Cite as

Cauchy Problem for Integrable Discrete Equations on Quad-Graphs

Article

Abstract

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 KdV 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 are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.

Keywords

Cauchy problem quad-graph discrete integrable equations 

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References

  1. 1.
    Adler, V. E.: Discrete equations on planar graphs, J. Phys. A: Math. Gen. 34 (2001), 10453–10460. MATHGoogle Scholar
  2. 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. MATHMathSciNetGoogle Scholar
  3. 3.
    Bianchi, L.: Vorlesungen über Differenzialgeometrie, Teubner, Leipzig, 1899. Google Scholar
  4. 4.
    Bobenko, A. I. and Suris, Yu. B.: Integrable systems on quad-graphs, Int. Math. Res. Notices 11 (2002), 573–611. MathSciNetGoogle Scholar
  5. 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. Google Scholar
  6. 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. MathSciNetGoogle Scholar
  7. 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]. MathSciNetGoogle Scholar
  8. 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. Google Scholar
  9. 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]. MathSciNetGoogle Scholar
  10. 10.
    Dynnikov, I. A. and Novikov, S. P.: Geometry of the triangle equation on two-manifolds, math-ph/0208041. Google Scholar
  11. 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. Google Scholar
  12. 12.
    King, A. and Schief, W.: Tetrahedra, octahedra and cubo-octahedra: Integrable geometry of multiratios, J. Phys. A: Math. Gen. 36 (2003), 785–802. MATHMathSciNetGoogle Scholar
  13. 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. MATHMathSciNetGoogle Scholar
  14. 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. MATHMathSciNetGoogle Scholar
  15. 15.
    Korepin, V. E.: Exactly solvable spin models for quasicrystals, JETP 92(3) (1987), 1082–1089. MathSciNetGoogle Scholar
  16. 16.
    Korepin, V. E.: Completely integrable models in quasicrystals, Comm. Math. Phys. 110(1) (1987), 157–171. MATHMathSciNetGoogle Scholar
  17. 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]. MATHMathSciNetGoogle Scholar
  18. 18.
    Miwa, T.: On Hirota’s difference equations, Proc. Japan Acad., Ser. A: Math. Sci. 58(1) (1982), 9–12. CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Nijhoff, F. W.: Lax pair for the Adler (lattice Krichever–Novikov) system, Phys. Lett. A 297(1–2) (2002), 49–58. MATHMathSciNetGoogle Scholar
  20. 20.
    Nijhoff, F. W. and Capel, H. W.: The discrete Korteweg–de Vries equation, Acta Appl. Math. 39 (1995), 133–158. MATHMathSciNetGoogle Scholar
  21. 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. MATHMathSciNetGoogle Scholar
  22. 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]. MathSciNetGoogle Scholar
  23. 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]. MathSciNetGoogle Scholar
  24. 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. MathSciNetGoogle Scholar
  25. 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. MATHMathSciNetGoogle Scholar
  26. 26.
    Senechal, M.: Quasicrystals and Geometry, Cambridge University Press, Cambridge, 1995. MATHGoogle Scholar
  27. 27.
    Veselov, A. P.: Yang–Baxter maps and integrable dynamics, Phys. Lett. A 314 (2003), 214–221. MATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Loughborough University, Loughborough, Leicestershire LE11 3TU, UK and Landau Institute for Theoretical PhysicsRussia

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