The B-quadrilateral lattice, its transformations and the algebro-geometric construction

Journal of Geometry and Physics

Volumn 57, Issue 4, 2007, Pages 1171-1192

  Doliwa, Adam ^a  

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

Author keywords

Darboux transformations; Discrete BKP equation; Integrable discrete geometry; Prym varieties

Indexed keywords

EID: 33845238251     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.geomphys.2006.09.010     Document Type: Article

References (47)

1. A.A. Akhmetshin, I.M. Krichever, Y.S. Volvovski, Discrete analogues of the Darboux-Egoroff metrics, Proc. Steklov Inst. Math. 225, 16-39
2. Białecki M., and Doliwa A. Algebro-geometric solution of the discrete KP equation over a finite field out of a hyperelliptic curve. Comm. Math. Phys. 253 (2005) 157-170
3. Bobenko A.I., and Pinkall U. Discrete isothermic surfaces. J. Reine Angew. Math. 475 (1996) 187-208
4. Bobenko A.I., and Suris Yu. Discrete differential geometry. Consistency as integrability. math.DG/0504358
5. Belokolos E.D., Bobenko A.I., Enol'skii V.Z., Its A.R., and Matveev V.B. Algebro-Geometric Approach to Nonlinear Integrable Equations (1994), Springer, Berlin
6. L.V. Bogdanov, B.G. Konopelchenko, Lattice and q-difference Darboux-Zakharov-Manakov systems via over(∂, ̄) method, J. Phys. A: Math. Gen. 28 L173-L178
7. Bogdanov L.V., and Konopelchenko B.G. Analytic-bilinear approach to integrable hiererchies II. Multicomponent KP and 2D Toda hiererchies. J. Math. Phys. 39 (1998) 4701-4728
8. Carroll G.D., and Speyer D. The cube recurrence. Electron. J. Combin. 11 1 (2004) 31 Research Paper 73, electronic
9. Cieśliński J., Doliwa A., and Santini P.M. The integrable discrete analogues of orthogonal coordinate systems are multidimensional circular lattices. Phys. Lett. A 235 (1997) 480-488
10. Coxeter H.S.M. Projective Geometry (1974), Springer, Berlin
11. Darboux G. Leçons sur les systémes orthogonaux et les coordonnées curvilignes (1910), Gauthier-Villars, Paris
12. Date E., Kashiwara M., Jimbo M., and Miwa T. Transformation groups for soliton equations. In: Jimbo M., and Miwa T. (Eds). Proceedings of RIMS Symposioum on Non-Linear Integrable Systems-Classical Theory and Quantum Theory (1983), World Science Publishing Co., Singapore 39-119
13. Date E., Jimbo M., Kashiwara M., and Miwa T. A new hierarchy of soliton equations of KP-type. Physica D 4 (1982) 343-365
14. Date E., Jimbo M., Kashiwara M., and Miwa T. Quasi-periodic solutions of the orthognal KP equation-Transformation groups for soliton equations V. Publ. RIMS, Kyoto Univ. 18 (1982) 1111-1119
15. Doliwa A. Quadratic reductions of quadrilateral lattices. J. Geom. Phys. 30 (1999) 169-186
16. A. Doliwa, Generalized isothermic lattice, nlin.SI/0611018
17. Doliwa A., and Santini P.M. Multidimensional quadrilateral lattices are integrable. Phys. Lett. A 233 (1997) 365-372
18. Doliwa A., and Santini P.M. The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice. J. Geom. Phys. 36 (2000) 60-102
19. Doliwa A., and Santini P.M. Integrable systems and discrete geometry. In: François J.P., Naber G., and Tsun T.S. (Eds). Encyclopedia of Mathematical Physics vol. III (2006), Elsevier 78-87
20. Doliwa A., Białecki M., and Klimczewski P. The Hirota equation over finite fields: Algebro-geometric approach and multisoliton solutions. J. Phys. A 36 (2003) 4827-4839
21. Doliwa A., Santini P.M., and Mañas M. Transformations of quadrilateral lattices. J. Math. Phys. 41 (2000) 944-990
22. Doliwa A., Grinevich P., Nieszporski M., and Santini P.M. Integrable lattices and their sub-lattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme. nlin.SI/0410046
23. Fay J.D. Theta Functions on Riemann Surfaces (1973), Springer, Berlin
24. Farkas H.M., and Kra I. Riemann Surfaces (1992), Springer, Berlin, New York
25. Fomin S., and Zelevinsky A. The Laurent phenomenon. Adv. Appl. Math. 28 (2002) 119-144
26. Hirota R. Soliton solutions to the BKP equations. I. The Pfaffian technique. J. Phys. Soc. Japan 58 (1989) 2285-2296
27. Kac V.G., and van de Leur J. The n-component KP hiererchy and representation theory. In: Fokas A.S., and Zakharov V.E. (Eds). Important Developments in Soliton Theory (1993), Springer, Berlin 302-343
28. Konopelchenko B.G., and Schief W.K. Trapezoidal discrete surfaces: geometry and integrability. J. Geom. Phys. 31 (1999) 75-95
29. Konopelchenko B.G., and Schief W.K. Reciprocal figures, graphical statics and inversive geometry of the Schwarzian BKP hierarchy. Stud. Appl. Math. 109 (2002) 89-124
30. Krichever I. A characterization of Prym varieties. math.AG/0506238
31. Liu Q.P., and Mañas M. Pfaffian form of Grammian determinant solutions of the BKP hierarchy. Chinese Ann. Math. Ser. A 23 (2002) 693-698. solv-int/9806004
32. Mañas M. Fundamental transformation for quadrilateral lattices: First potentials and τ-functions, symmetric and pseudo-Egorov reductions. J. Phys. A 34 (2001) 10413-10421
33. Mañas M., Doliwa A., and Santini P.M. Darboux transformations for multidimensional quadrilateral lattices. I. Phys. Lett. A 232 (1997) 99-105
34. Miwa T. On Hirota's difference equations. Proc. Japan Acad. 58 (1982) 9-12
35. Möbius F.A. Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um- und eingeschrieben zugleich heissen?. J. Reine Angew. Math. 3 (1828) 273-278
36. Muir T. A Treatise on The Theory of Determinants. revised and enlarged by W. H. Metzler (1960), Dover Publ., New York
37. Mumford D. Prym varieties. I. Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers) (1974), Academic Press, New York 325-350
38. Nimmo J.J.C., and Schief W.K. Superposition principles associated with the Moutard transformation. An integrable discretisation of a (2+1)-dimensional sine-Gordon system. Proc. R. Soc. Lond. Ser. A 453 (1997) 255-279
39. Pedoe D. Geometry, A Comprehensive Course (1988), Dover Publications, New York
40. Propp J. The many faces of alternating-sign matrices. Discrete Math. Theoret. Comput. Sci. Proc. AA (DM-CCG) (2001) 43-58
41. Proskuryakov I.V. Problems in Linear Algebra (1978), Mir Publishers, Moscow
42. Sauer R. Differenzengeometrie (1970), Springer, Berlin
43. Schief W.K. Lattice geometry of the discrete Darboux, KP, BKP and CKP equations. Menelaus' and Carnot's theorems. J. Nonlinear Math. Phys. 10 Suppl. 2 (2003) 194-208
44. Shiota T. Prym varieties and soliton equations. In: Kac V.G. (Ed). Infinite Dimensional Lie Algebras and Groups (1989), World Scientific, Singapore 407-448
45. Taimanov I.A. Prym varieties of branch covers and nonlinear equations. Mat. Sb. 181 (1990) 934-950
46. Tsujimoto S., and Hirota R. Pfaffian representation of solutions to the discrete BKP hierarchy in bilinear form. J. Phys. Soc. Japan 66 (1997) 2797-2806
47. Veselov A.P., and Novikov S.P. Finite-gap two-dimensional potential Schrödinger operators: Explicit formulae and evolution equations. Dokl. AN USSR 279 (1984) 20-24