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Special Solutions of Discrete Integrable Systems

Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 644)

Abstract

Hierarchies of discrete soliton equations are constructed in bilinear form as a consequence of the algebraic identities satisfied by determinants and Pfaffians. Difference formulas for determinants and Pfaffians are derived from the discrete linear dispersion relations satisfied by their elements. For completeness, we first summarize the main algebraic properties of determinants and Pfaffians.

Keywords

Bilinear Form Special Solution Darboux Transformation Soliton Equation Expansion Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1. T. Muir: A Treatise on the Theory of Determinants (Macmillan and Co., London 1882).Google Scholar
  2. 2. J. Satsuma: J. Phys. Soc. Jpn. 46, 359 (1979).Google Scholar
  3. 3. N. C. Freeman, J. J. C. Nimmo: Phys. Lett. 95A, 1 (1983).CrossRefGoogle Scholar
  4. 4. N. C. Freeman: IMA J. Appl. Math. 32, 125 (1984).Google Scholar
  5. 5. R. Hirota: J. Phys. Soc. Jpn. 55, 2137 (1986).Google Scholar
  6. 6. J. J. C. Nimmo: Symmetric functions and the KP hierarchy. In: Nonlinear Evolutions, ed. by J. J. P. Leon (World Scientific, Singapore 1988) pp 245–261.Google Scholar
  7. 7. A. Nakamura: J. Phys. Soc. Jpn. 58, 412 (1989).Google Scholar
  8. 8. S. Miyake, Y. Ohta, J. Satsuma: J. Phys. Soc. Jpn. 59, 48 (1990).Google Scholar
  9. 9. Y. Ohta, R. Hirota, S. Tsujimoto, T. Imai: J. Phys. Soc. Jpn. 62, 1872 (1993).zbMATHGoogle Scholar
  10. 10. R. Hirota, Y. Ohta: J. Phys. Soc. Jpn. 60, 798 (1991).Google Scholar
  11. 11. C. R. Gilson, J. J. C. Nimmo: Theo. Math. Phys. 128, 870 (2001).CrossRefGoogle Scholar
  12. 12. C. R. Gilson, J. J. C. Nimmo, S. Tsujimoto: J. Phys. A 34, 10569 (2001).CrossRefGoogle Scholar
  13. 13. R. Hirota: J. Phys. Soc. Jpn. 43, 1424, 2074, 2079 (1977); ibid 45, 321 (1978).Google Scholar
  14. 14. E. Date, M. Jimbo, T. Miwa: J. Phys. Soc. Jpn. 51, 4116, 4125 (1982); ibid 52, 388, 761, 766 (1983).Google Scholar
  15. 15. M. Sato: RIMS Kokyuroku 439, 30 (1981).Google Scholar
  16. 16. M. Sato, Y. Sato: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, In: Nonlinear Partial Differential Equations in Applied Science, ed. by H. Fujita, P. D. Lax, G. Strang (Kinokuniya, North-Holland, Tokyo 1983) pp 259–271.Google Scholar
  17. 17. M. Sato: Soliton Equations and the Universal Grassmann Manifolds, notes by M. Noumi (Sophia Kokyuroku in Mathematics 18, Dept. Math. Sophia Univ., Tokyo 1984).Google Scholar
  18. 18. Y. Ohta, J. Satsuma, D. Takahashi, T. Tokihiro: Prog. Theo. Phys. Suppl. 94, 210 (1988).MathSciNetGoogle Scholar
  19. 19. J. Satsuma, R. Willox: Sato theory and transformation group theory approach to integrable systems. in this volume.Google Scholar
  20. 20. E. Date, M. Kashiwara, M. Jimbo, T. Miwa: Transformation groups for soliton equations. In: Non-linear Integrable Systems — Classical Theory and Quantum Theory, ed. by M. Jimbo, T. Miwa (World Scientific, Singapore 1983) pp 39-119.Google Scholar
  21. 21. M. Jimbo, T. Miwa: Publ. RIMS, Kyoto Univ. 19, 943 (1983).Google Scholar
  22. 22. T. Miwa, M. Jimbo, E. Date: Solitons. Differential Equations, Symmetries and Infinite-Dimensional Algebras, translated by M. Reid (Cambridge Tracts in Mathematics 135, Cambridge University Press, Cambridge 2000).Google Scholar
  23. 23. T. Miwa: Proc. Jpn. Acad. 58A, 9 (1982).Google Scholar
  24. 24. H. Harada: J. Phys. Soc. Jpn. 54, 4507 (1985); ibid 56, 3847 (1987).Google Scholar
  25. 25. J. J. C. Nimmo: Inverse Problems 8, 219 (1992); J. Phys. A 30, 8693 (1997).CrossRefGoogle Scholar
  26. 26. S. Kakei: J. Phys. Soc. Jpn. 68, 2875 (1999).Google Scholar
  27. 27. M. Adler, T. Shiota, P. van Moerbeke: Math. Ann. 322, 423 (2002).CrossRefGoogle Scholar

Authors and Affiliations

  • Y. Ohta
    • 1
  1. 1.Department of Mathematics, Kobe University, Rokko, Kobe 657-8501Japan

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