The symmetric, D -invariant and Egorov reductions of the quadrilateral lattice

Journal of Geometry and Physics

Volumn 36, Issue 1-2, 2000, Pages 60-102

  Doliwa, Adam ^a   Santini, Paolo Maria ^b,c  

^a UNIVERSITY OF WARSAW   (Poland)

^b INFN SEZIONE DI ROMA   (Italy)

^c SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

02.40.Hw; 04.60.Nc; 51M30; 52C07; 53A20; 58F07; Differential geometry; Discrete geometry; Integrable systems

Indexed keywords

EID: 0002659187     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(00)00011-5     Document Type: Article

References (40)

1. A.A. Akhmetshin, I.M. Krichever, Y.S. Volvovski, Discrete Analogs of the Darboux-Egorov Metrics. hep-th/9905168.
2. L. Bianchi, Lezioni di Geometria Differenziale, 3rd Edition, Zanichelli, Bologna, 1924.
3. A. Bobenko, Discrete conformal maps and surfaces, in: P. Clarkson, F. Nijhoff (Eds.), Symmetries and Integrability of Difference Equations, Cambridge University Press, Cambridge, 1999, pp. 97-108.
4. Bobenko A., Pinkall U. Discrete isothermic surfaces. J. Reine Angew. Math. 475:1996;187-208.
5. Bogdanov L.V., Konopelchenko B.G. Lattice and q-difference Darboux-Zakharov-Manakov systems via. ∂̄ method J. Phys. A. 28:1995;L173-L178.
6. Bogdanov L.V., Konopelchenko B.G. Analytic-bilinear approach to integrable hierarchies II. Multicomponent KP and 2D Toda lattice hierarchies. J. Math. Phys. 39:1998;4701-4728.
7. Bogdanov L.V., Manakov S.V. The nonlocal. ∂̄ -problem and (2+1) -dimensional soliton equations J. Phys. A. 21:1988;L537-L544.
8. Cieśliński J., Doliwa A., Santini P.M. The integrable discrete analogues of orthogonal coordinate systems are multidimensional circular lattices. Phys. Lett. A. 235:1997;480-488.
9. G. Darboux, Leçons sur les Systémes Orthogonaux et les Coordonnées Curvilignes, 2nd Edition, Gauthier-Villars, Paris, 1910.
10. E. Date, M. Kashiwara, M. Jimbo, T. Miwa, Transformation groups for soliton equations, in: M. Jimbo, T. Miwa (Eds.), Nonlinear Integrable Systems - Classical Theory and Quantum Theory, Proceedings of the RIMS Symposium, World Scientific, Singapore, 1983, pp. 39-119.
11. Doliwa A. Geometric discretisation of the Toda system. Phys. Lett. A. 234:1997;187-192.
12. Doliwa A. Quadratic reductions of quadrilateral lattices. J. Geom. Phys. 30:1999;169-186.
13. Doliwa A., Santini P.M. Multidimensional quadrilateral lattices are integrable. Phys. Lett. A. 233:1997;365-372.
14. A. Doliwa, P.M. Santini, in: Talks given at the Workshop on Symmetries and Integrability of Difference Equations, Sabaudia, May 1998.
15. A. Doliwa, P.M. Santini, in: Talks given at the Workshop on Integrable Systems: Solutions and Transformations, Alicante, June 1998.
16. Doliwa A., Manakov S.V., Santini P.M. ∂̄ -reductions of the multidimensional quadrilateral lattice: the multidimensional circular lattice Commun. Math. Phys. 196:1998;1-18.
17. Doliwa A., Mañas M., Martínez Alonso L., Medina E., Santini P.M. Charged free fermions, vertex operators and transformation theory of conjugate nets. J. Phys. A. 32:1999;1197-1216.
18. Doliwa A., Santini P.M., Mañas M. Transformations of quadrilateral lattices. J. Math. Phys. 41:2000;944-990.
19. Dubrovin B. Integrable systems in topological field theory. Nuclear Phys. B. 379:1992;627-689.
20. Dubrovin B.A., Novikov S.P. Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Witham averaging method. Dokl. Akad. Nauk SSSR. 270:1983;781-785.
21. Ferapontov E.V. Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type. Funct. Anal. Appl. 25:1991;195-204.
22. j and Segal-Wilson τ -function, in: A.A. Belavin, A.U. Klimyk, A.B. Zamolodchikov (Eds.), Problems in Modern QFT, Springer, Berlin, 1989, pp. 86-106.
23. Grinevich P.G., Orlov A.Yu. Variations of the complex structure of Riemann surfaces by vector fields on the circle and objects in KP theory. The Krichever-Novikov problem of the actions by the Baker-Akhiezer function. Funct. Anal. Appl. 24:1990;61-63.
24. Grundland A.M., Żelazny R. Simple waves in quasilinear hyperbolic systems. I. Theory of simple waves and simple states. Examples of applications. II. Riemann invariants for the problem of simple wave interactions. J. Math. Phys. 24:1983;2305-2328.
25. Konopelchenko B.G., Schief W.K. Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. Proc. Roy. Soc. London A. 454:1998;3075-3104.
26. I.M. Krichever, Seminar, University of Rome, La Sapienza, June 1998.
27. Mañas M., Doliwa A., Santini P.M. Darboux transformations for multidimensional quadrilateral lattices I. Phys. Lett. A. 232:1997;99-105.
28. R.R. Martin, J. de Pont, T.J. Sharrock, Cyclic surfaces in computer aided design, in: J.A. Gregory (Ed.), The Mathematics of Surfaces, Clarendon Press, Oxford, 1986, pp. 253-268.
29. Nimmo J.J.C., Schief W.K. Superposition principles associated with the Moutard transformation. An integrable discretisation of a. (2+1) -dimensional Sine-Gordon system Proc. Roy. Soc. London A. 453:1997;255-279.
30. A.W. Nutbourne, The solution of a frame matching equation, in: J.A. Gregory (Ed.), The Mathematics of Surfaces, Clarendon Press, Oxford, 1986, pp. 233-252.
31. P.M. Santini, in: Talks given at the Workshop on Symmetry and Perturbation Theory, Rome, December 1998.
32. R. Sauer, Differenzengeometrie, Springer, Berlin, 1970.
33. W.K. Schief, in: Talk given at the Workshop: Nonlinear Systems, Solitons and Geometry, Oberwolfach, October 1997.
34. W.K. Schief, private communication, February 1998.
35. W.K. Schief, in: The Workshop on Symmetries and Integrability of Difference Equations, Sabaudia, May 1998, private communication.
36. W.K. Schief, private communication, October 1999.
37. S.P. Tsarev, Classical differential geometry and integrability of systems of hydrodynamic type, in: P.A. Clarkson (Ed.), Application of Analytic and Geometric Methods to Nonlinear Differential Equations, Kluwer Academic Publishers, Dordrecht, 1993, pp. 241-249.
38. V.E. Zakharov, On the dressing method, in: P.C. Sabatier (Ed.), Inverse Method in Action, Springer, Berlin, 1990, pp. 602-623.
39. Zakharov V.E., Manakov S.V. Construction of multidimensional nonlinear integrable systems and their solutions. Funct. Anal. Appl. 19:1985;89.
40. Zakharov V.E., Manakov S.V. Reductions in systems integrated by the method of the inverse scattering problem. Dokl. Math. 57:1998;471-474.