Discrete Levy transformations and Casorati determinant solutions of quadrilateral lattices

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Sequences of discrete Levy and adjoint Levy transformations for multidimensional quadrilateral lattices are studied. After a suitable number of iterations we show how all the relevant geometrical features of the transformed quadrilateral lattice can be expressed in terms of multi-Casorati determinants. As an example we dress the Cartesian lattice.
©1998 Published by Elsevier Science B.V.
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[1] L. V. Bogdanov and B. G. Konopelchenko, J. Phys. A: Math. & Gen. 28 (1995) L173. [2] J. Cieśliński, A. Doliwa and P. M. Santini, The Integrable Discrete Analogous of Orthogonal Coordinate Systems are Multidimensional Circular Lattices, to appear in Phys. Lett. A (1997). [3] G. Darboux, Le¸cons sur la théorie générale des surfaces IV, Liv. VIII, Chap. XII, Gauthier – Villars, Paris (1896). Reprinted by Chelsea Publishing Company, New York (1972). [4] A. Doliwa, S. V. Manakov and P. M. Santini, ¯∂-Reductions of the Multidimensional Quadrilateral Lattice I: The Multidimensional Circular Lattice, to appear (1997). [5] A. Doliwa, M. Mañas, L. Martínez Alonso, E. Medina and P. M. Santini, The Miwa Transformation and τ -Functions for Quadrilateral Lattices. Geometrical Interpretation, to appear (1997). [6] A. Doliwa and P. M. Santini, Multidimesional Quadrilateral Lattices are Integrable, to appear in Phys. Lett. A (1997). [7] A. Doliwa, P. M. Santini and M. Mañas, Transformations for Quadrilateral Lattices, to appear (1997). [8] L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Co., Boston (1909). [9] N.C. Freeman and J.J.C. Nimmo, Phys. Lett. 95 A (1983) 1; Phys. Lett. 95 A (1983) 4; J. J. C. Nimmo, Phys. Lett. 99 A (1983) 281; N.C. Freeman, IMA J. Appl. Math. 32 (1984) 125. [10] E. S. Hammond, Ann. Math. 22 (1920) 238. [11] B. G. Konopelchenko and W. K. Shief, Lamé and Zakharov-Manakov systems: Combescure, Darboux and Bäcklund transformations, Preprint AM93/9, UNSW (1993). [12] L. Levy, J. l’École Polytecnique 56 (1886) 63. [13] Q. P. Liu and M. Mañas, Sequences of Levy transformations and multiWro´nski determinant solutions of the Darboux system, dg-ga/9707013 (1997). [14] M. Mañas, A. Doliwa and P. M. Santini, Phys. Lett. 232 A (1997) 365. [15] Y. Ohta, R. Hirota, S. Tsujimoto and T. Imai, J. Phys. Soc. Japan 62 (1993) 1872. [16] O. Schreier and E. Sperner, Introduction to Modern Algebra and Matrix Theory, Chelsea Publishing Company, New York (1951).