Publication: Generating quadrilateral and circular lattices in KP theory
Loading...
Official URL
Full text at PDC
Publication Date
1999-11-08
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
The bilinear equations of the N-component KP and BKP hierarchies and a corresponding extended Miwa transformation allow us to generate quadrilateral and circular lattices from conjugate and orthogonal nets, respectively. The main geometrical objects are expressed in terms of Baker functions.
Description
©Physics letters A.
UCM subjects
Unesco subjects
Keywords
Citation
[1] L. Bianchi, Lezioni di Geometria Differenziale, 3-a ed., Zanichelli, Bologna (1924).
[2] A. Bobenko, in Symmetries and Integrability of Difference Equations II, P. Clarkson and F. Nijhoff (eds.), Cambridge University Press, Cambridge (1997).
[3] L. V. Bogdanov and B. G. Konopelchenko, J. Math. Phys. 39 (1998) 4701.
[4] J. Ciéslínski, A. Doliwa and P. M. Santini, Phys. Lett. A235 (1997) 480.
[5] G. Darboux, Le¸cons sur la théorie générale des surfaces IV, GauthierVillars, Paris (1896) Reprinted by Chelsea Publishing Company, New York (1972).
[6] G. Darboux, Le¸cons sur les systèmes orthogonaux et les coordenées curvilignes (deuxième édition), Gauthier-Villars, Paris (1910) (the first edition was in 1897). Reprinted by Éditions Jacques Gabay, Sceaux (1993).
[7] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, J. Phys. Soc. Japan 50 (1981) 3806.
[8] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Physica D4 (1982) 343.
[9] E. Date, M. Jimbo and T. Miwa, J. Phys. Soc. Japan 51 (1982) 4116, 4125; 52 (1983) 388, 761, 766.
[10] M. A. Demoulin, Comp. Rend. Acad. Sci. Paris 150 (1910) 156.
[11] A. Doliwa, Quadratic reductions of quadrilateral lattices, solv-int/9802011.
[12] A. Doliwa, S. V. Manakov and P. M. Santini, ¯∂-Reductions of the Multidimensional Quadrilateral Lattice I: the Multidimensional Circular Lattice. Commun. Math. Phys. (in press).
[13] A. Doliwa, M. Mañas, L. Martínez Alonso, E. Medina and P. M. Santini, Charged Free Fermions, Vertex Operators and Classical Theory of Conjugate Nets, solv-int/9803015.
[14] A. Doliwa and P. M. Santini, Phys. Lett. A 233 (1997) 365.
[15] A. Doliwa, P. M. Santini and M. Mañas, Transformations for Quadrilateral Lattices, solv- int/9712017.
[16] L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Co., Boston (1909).
[17] L. P. Eisenhart, Trans. Amer. Math. Soc. 18 (1917) 111.
[18] L. P. Eisenhart, Transformations of Surfaces, Princeton University Press, Princeton (1923). Reprinted by Chelsea Publishing Company, New York (1962).
[19] E. I. Ganzha, On completeness of the Ribaucour transformations for triply orthogonal curvilinear coordinate systems in R 3 , solv-int/9606002.
[20] E. I. Ganzha and S. P. Tsarev, On superposition of the autoBäcklund transformations for (2+1)-dimensional integrable systems, solv-int/9606003.
[21] R. Hirota, J. Phys. Soc. Japan 50 (1981) 3785.
[22] H. Jonas, Berl. Math. Ges. Ber. Sitzungsber. 14 (1915) 96.
[23] V. G. Kac and J. van de Leur, in Important Developments in Soliton Theory, edited by A. S. Fokas and V. E. Zakharov (Springer, Berlin, 1993) p. 302.
[24] B. G. Konopelchenko and W. K. Schief, Three-dimensional Integrable Lattices in Euclidean Space: Conjugacy and Orthogonality, Preprint University of New South Wales (1997).
[25] G. Lamé, Le¸cons sur la théorie des coordenées curvilignes et leurs diverses applications, Mallet-Bachalier, Paris (1859).
[26] L. Lévy, J. l’Ecole Polytecnique 56 (1886) 63.
[27] Q. P. Liu and M. Mañas, Phys. Lett. A239 (1998) 159.
[28] Q. P. Liu and M. Mañas, J. Phys. A: Math. & Gen. 31 (1998) L193.
[29] Q. P. Liu and M. Mañas, Superposition Formulae for the Discrete Ribaucour Transformations of Circular Lattices, Phys. Lett. A (in press).
[30] M. Mañas, A. Doliwa and P. M. Santini, Phys. Lett. A232 (1997) 365.
[31] M. Mañas and L. Martinez Alonso, From Ramond Fermions to Lamé Equations for Orthogonal Curvilinear Coordinates, Phys. Lett. B. (in press), solv-int/9805010.
[32] R. R. Martin, in The Mathematics of Surfaces, J. Gregory (ed.), Claredon Press, Oxford (1986).
[33] T. Miwa, Proc. Japan Acad. A58 (1982) 9.
[34] A. W. Nutborne, in The Mathematics of Surfaces, J. Gregory (ed.), Claredon Press, Oxford (1986).
[35] A. Ribaucour, Comp. Rend. Acad. Sci. Paris 74 (1872) 1489.
[36] M. Sato, RIMS Kokyuroku 439 (1981) 30; in Theta Functions–Bowdoin 1987. Part 1 Proc. Sympos. Pure Maths. 49, part 1, 51, AMS (1989) Providence.
[37] R. Sauer, Differenzengeometrie, Springer-Verlag, Berlin (1970)
[38] G. Segal and G. Wilson, Publ. Math. I. H. E. S. 61, 5, 1985.
[39] V. E. Zakharov, On Integrability of the Equations Describing NOrthogonal Curvilinear Coordinate Systems and Hamiltonian Integrable Systems of Hydrodynamic Type I: Integration of the Lam´e Equations Preprint (1996). [40] V. E. Zakharov and S. V. Manakov, Func. Anal. Appl. 19 (1985) 11.