Algebro-geometric solution of the discrete KP equation over a finite field out of a hyperelliptic curve

Communications in Mathematical Physics

Volumn 253, Issue 1, 2005, Pages 157-170

  Białecki, Mariusz ^a   Doliwa, Adam ^b  

^a INSTITUTE OF GEOPHYSICS   (Poland)

^b UNIVERSITY OF WARMIA AND MAZURY IN OLSZTYN   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 11244264164     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-004-1207-3     Document Type: Article

References (29)

1. Belokolos, E.D., Bobenko, A.I., Enol'skii, V.Z., Its, A.R., Matveev, V.B.: Algebro-geometric approach to nonlinear integrable equations. Berlin: Springer-Verlag, 1994
2. Białecki, M.: Methods of algebraic geometry over finite fields in construction of integrable cellular automata. PhD dissertation, Warsaw University, Institute of Theoretical Physics, 2003
3. Białecki, M., Doliwa, A.: The discrete KP and KdV equations over finite fields. Theor. Math. Phys. 137, 1412-1418 (2003)
4. Bobenko, A., Bordemann, M., Gunn, Ch., Pinkall, U., On two integrable cellular automata. Commun. Math. Phys. 158, 127-134 (1993)
5. Bruschi, M., Santini, P.M.: Cellular automata in 1+1, 2+1 and 3+1 dimensions, constants of motion and coherent structures. Physica D 70, 185-209 (1994)
6. Cornell, G., Silverman, J.H. (eds.): Arithmetic geometry. New York: Springer-Verlag, 1986
7. Doliwa, A., Białecki, M., Klimczewski, P.: The Hirota equation over finite fields: algebro-geometric approach and multisoliton solutions. J. Phys. A 36, 4827-4839 (2003)
8. Fokas, A.S., Papadopoulou, E.P., Saridakis, Y.G.: Soliton cellular automata. Physica D 41, 297-321 (1990)
9. Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: John Wiley and Sons, 1978
10. Hartshorne, R.: Algebraic geometry. New York: Springer-Verlag. 1977
11. Hirota, R.: Nonlinear partial difference equations. III. Discrete sine-Gordon equation. J. Phys. Soc. Jpn. 43, 2079-2086 (1977)
12. Hirota, R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50, 3785-3791 (1981)
13. Koblitz, N.: Algebraic aspects of cryptography. Berlin: Springer-Verlag, 1998
14. Krichever, I.M.: Algebraic curves and non-linear difference equations. Usp. Mat. Nauk 33, 215-216 (1978)
15. Krichever, I.M., Wiegmann, P., Zabrodin, A.: Elliptic solutions to difference non-linear equations and related many body problems. Commun. Math. Phys. 193, 373-396 (1998)
16. Lang, S.: Abelian varieties. New York: Interscience Publishers Inc. 1958
17. Lang. S.: Algebra. Reading, MA: Addison-Wesley, 1970
18. Matsukidaira, J., Satsuma, J., Takahashi, D., Tokihiro, T., Torii, M.: Toda-type cellular automaton and its N-soliton solution. Phys. Lett. A 225, 287-295 (1997)
19. Menezes, A.J., Wu, Y.H., Zuccherato, R.J.: An elementary introduction to hyperelliptic curves. Appendix in [13], pp. 151-178
20. Milne, J.S.: Jacobian varieties. Chapter 16 in [6], pp. 167-212
21. Moreno, C.: Algebraic curves over finite fields. Cambridge: University Press, 1991
22. Mumford, D.: An algebra-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation, and related nonlinear equations. Proceedings of the International Symposium on Algebraic Geometry (M. Nagata, ed.), Kinokuniya, Tokyo, 1978 pp. 115-153
23. von Neumann, J.: The general and logical theory of automata. In: The collected works of John von Neumann (A.W. Taub, ed.), Vol. 5, New York: Pergamon Press, 1963
24. Shafarevich, I.: Basic algebraic geometry. Heidelberg: Springer-Verlag, 1974
25. Stichtenoth, H.: Algebraic function fields and codes. Berlin: Springer-Verlag, 1993
26. Takahashi, D., Satsuma, J.: A soliton cellular automaton. J. Phys. Soc. Jpn. 59, 3514-3519 (1990)
27. 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)
28. Ulam, S.: Random processes and transformations. In: Proceedings of the International Congress of Mathematicians, Cambridge, MA, 30 August-6 September 1950 (P. A. Smith, O. Zariski, eds.), Providence, RI: AMS, 1952, pp. 264-275
29. Wolfram, S.: Theory and application of cellular automata. Singapore: World Scientific, 1986