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Ultradiscrete permanent solution to the ultradiscrete Kadomtsev–Petviashvili equation

  • Hidetomo NagaiEmail author
Original Paper Area 1

Abstract

We propose an ultradiscrete permanent solution to the ultradiscrete Kadomtsev–Petviashvili (KP) equation. The ultradiscrete permanent is an ultradiscrete analogue of the usual permanent. The elements on this ultradiscrete permanent solution are required some additional relations other than the ultradiscrete dispersion relation. We confirm the solution satisfying these relations and propose some explicit examples of the solution.

Keywords

Soliton equation Ultradiscrete system Kadomtsev–Petviashvili equation Ultradiscrete Permanent 

Mathematics Subject Classification

35C08 39C08 

Notes

Acknowledgements

The author is grateful to Professor Daisuke Takahashi for helpful advice. The author also grateful Keisuke Mizuki for valuable discussions.

References

  1. 1.
    Hirota, R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50, 3785–91 (1981)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Miwa, T.: On Hirota’s difference equations. Proc. Jpn. Acad. 58, 9–12 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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. Jpn. 62, 1872–1886 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Tokihiro, T., Takahashi, D., Matsukidaira, J., Satsuma, J.: From Soliton Equations to Integrable Cellular Automata through a Limiting Procedure. Phys. Rev. Lett. 76, 3247–3250 (1996)CrossRefGoogle Scholar
  5. 5.
    Takahashi, D., Hirota, R.: Ultradiscrete soliton solution of permanent type. J. Phys. Soc. Jpn. 76, 104007–104012 (2007)CrossRefGoogle Scholar
  6. 6.
    Nagai, H., Takahashi, D.: Ultradiscrete Plücker relation specialized for soliton solutions. J. Phys. A Math. Theor. 44, 095202 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Nagai, H.: A new expression of a soliton solution to the ultradiscrete Toda equation. J. Phys. A Math. Theor. 41, 235204 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nakamura, S.: A periodic phase soliton of the ultradiscrete hungry Lotka-Volterra equation. J. Phys. A Math. Theor. 42, 495204 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTokai UniversityHiratsukaJapan

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