Nets, lattices, and multicomponent KP hierarchies
Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 2, pp. 272-283 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We review some recent discoveries concerning the relation of geometric nets and lattices to the theory of the KP hierarchy.
@article{TMF_2000_122_2_a9,
     author = {M. Ma\~nas},
     title = {Nets, lattices, and multicomponent {KP} hierarchies},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {272--283},
     year = {2000},
     volume = {122},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a9/}
}
TY  - JOUR
AU  - M. Mañas
TI  - Nets, lattices, and multicomponent KP hierarchies
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2000
SP  - 272
EP  - 283
VL  - 122
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a9/
LA  - ru
ID  - TMF_2000_122_2_a9
ER  - 
%0 Journal Article
%A M. Mañas
%T Nets, lattices, and multicomponent KP hierarchies
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2000
%P 272-283
%V 122
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a9/
%G ru
%F TMF_2000_122_2_a9
M. Mañas. Nets, lattices, and multicomponent KP hierarchies. Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 2, pp. 272-283. http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a9/

[1] A. Doliwa, M. Mañas, L. Martínez Alonso, E. Medina, P. M. Santini, Charged Free Fermions, Vertex Operators and Classical Theory of Conjugate Nets, E-print solv-int/9803015 | MR

[2] M. Mañas, L. Martinez Alonso, Phys. Lett. B, 436 (1998), 316 | DOI | MR

[3] A. Doliwa, M. Mañas, L. Martínez Alonso, Generating Quadrilateral and Circular Lattices in KP Theory, E-print solv-int/9810012

[4] L. P. Eisenhart, Transformations of Surfaces, Princeton University Press, Princeton, 1923 ; Reprinted, Chelsea Publishing Company, New York, 1962 | Zbl

[5] G. Darboux, Leçons sur la théorie générale des surfaces, Gauthier-Villars, Paris, 1896 ; Reprinted, Chelsea Publishing Company, New York, 1972 | MR | Zbl

[6] L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Co., Boston, 1909 | MR | Zbl

[7] E. P. Lane, Projective Differential Geometry of Curves and Surfaces, University of Chicago Press, Chicago, 1932 | MR | Zbl

[8] D. J. Struik, Lectures on Classical Differential Geometry, Dover, New York, 1961 | MR

[9] A. Doliwa, P. M. Santini, M. Mañas, Transformations for Quadrilateral Lattices, E-print solv-int/9712017

[10] L. Lévy, J. l'École Polytecnique, 56 (1886), 63 | Zbl

[11] H. Jonas, Berl. Math. Ges. Ber. Sitzungsber., 14 (1915), 96 | Zbl

[12] L. P. Eisenhart, Trans. Amer. Math. Soc., 18 (1917), 111 | DOI | MR

[13] A. Doliwa, S. V. Manakov, P. M. Santini, Commun. Math. Phys., 196 (1998), 1 | DOI | MR | Zbl

[14] G. Lamé, Leçons sur la théorie des coordenées curvilignes et leurs diverses applications, Mallet-Bachalier, Paris, 1859

[15] G. Darboux, Leçons sur les systèmes orthogonaux et les coordenées curvilignes (deuxième édition), Gauthier-Villars, Paris, 1910 ; Reprinted, Éditions Jacques Gabay, Sceaux, 1993, the first edition was in 1897 | MR

[16] L. Bianchi, Lezioni di Geometria Differenziale, 3-a ed., Zanichelli, Bologna, 1924 | Zbl

[17] A. Ribaucour, Compt. Rend. Acad. Sci. Paris, 74 (1872), 1489

[18] R. Sauer, Differenzengeometrie, Springer-Verlag, Berlin, 1970 | MR

[19] A. Doliwa, P. M. Santini, Phys. Lett. A, 233 (1997), 365 | DOI | MR | Zbl

[20] R. R. Martin, The Mathematics of Surfaces, ed. J. Gregory, Claredon Press, Oxford, 1986 | MR | Zbl

[21] A. Bobenko, Symmetries and Integrability of Difference Equations, V. II, eds. P. Clarkson and F. Nijhoff, Cambridge University Press, Cambridge, 1998 ; A. Bobenko, U. Pinkall, J. Diff. Geom., 43 (1996), 527 ; J. Reine Angew. Math., 475 (1996), 187 ; A. Bobenko, W. K. Schief, Discrete Integrable Geometry and Physics, eds. A. Bobenko, R. Seiler, Oxford University Press, Oxford, 1998 | MR | DOI | MR | Zbl | MR | Zbl | MR

[22] J. Cieśliński, A. Doliwa, P. M. Santini, Phys. Lett. A, 235 (1997), 480 | DOI | MR | Zbl

[23] B. G. Konopelchenko, W. K. Schief, Three-dimensional Integrable Lattices in Euclidean Space: Conjugacy and Orthogonality, Preprint University of New South Wales, 1997 | MR

[24] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, J. Phys. Soc. Japan, 50 (1981), 3806 | DOI | MR | Zbl

[25] V. G. Kac, J. W. van de Leur, “The $n$-component KP hierarchy and representation theory”, Important Developments in Soliton Theory, eds. A. S. Fokas, V. E. Zakharov, Springer, Berlin, 1993, 302 | DOI | MR | Zbl

[26] L. V. Bogdanov, B. G. Konopelchenko, J. Math. Phys., 39 (1998), 4701 | DOI | MR | Zbl

[27] M. Jimbo,T. Miwa, Physica D, 1981, 407 ; L. Schlesinger, J. Reine Angew. Math., 14 (1912), 96 | DOI | MR | Zbl

[28] M. Sato, RIMS Kokyuroku, 439 (1981), 30 ; “The KP hierarchy and infinite-dimensional Grassmann manifolds”, Theta Functions, Part 1, Bowdoin, 1987; Part 1, Proc. Sympos. Pure Math., 49, AMS, Providence, RI, 1989, 51 | Zbl | DOI

[29] G. Segal, G. Wilson, Publ. Math. IHES, 61 (1985), 5 | DOI | Zbl

[30] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Physica D, 4 (1982), 343 | DOI | MR | Zbl

[31] T. Miwa, Proc. Japan Acad. A, 58 (1982), 9 ; E. Date, M. Jimbo, T. Miwa, J. Phys. Soc. Japan, 51 (1982), 4116 ; 4125 ; 52 (1983), 388 ; 761 ; 766 | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl | Zbl