Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices. We finally interpret these results within the formalism.
REFERENCES
1.
G. Darboux, Leçons Sur la Théorie Générale des Surfaces I–IV (Gauthier-Villars, Paris, 1887–1896).
2.
G. Darboux, Leçons Sur les Systémes Orthogonaux et les Coordonnées Curvilignes (Gauthier-Villars, Paris, 1910).
3.
L. P. Eisenhart, Transformations of Surfaces (Princeton University Press, Princeton, 1923).
4.
G. Lamé, Leçons Sur les Coordonnées Curvilignes et Leurs Diverses Applications (Mallet-Bachalier, Paris, 1859).
5.
V. E.
Zakharov
and S. V.
Manakov
, “Construction of multidimensional nonlinear integrable systems and their solutions
,” Funct. Anal. Appl.
19
, 89
(1985
).6.
V. E.
Zakharov
, “On integrability of the equations describing N-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. Part 1. Integration of the Lamé equations
,” Duke Math. J.
94
, 103
–139
(1998
).7.
V. E.
Zakharov
and S. V.
Manakov
, “Reductions in systems integrated by the method of the inverse scattering problem
,” Dokl. Akad. Nauk (Math.)
57
, 471
–474
(1998
).8.
A. Sym, “Soliton surfaces and their applications (soliton geometry from spectral problems),” in Geometric Aspects of the Einstein Equations and Integrable Systems, Lecture Notes in Physics 239 (Springer-Verlag, Berlin, 1985), pp. 154–231.
9.
A. Bobenko, “Surfaces in terms of 2 by 2 matrices. Old and new integrable cases,” in Harmonic Maps and Integrable Systems, edited by A. Fordy and J. Wood (Vieweg, Braunschweig, 1994).
10.
A.
Bobenko
and U.
Pinkall
, “Discrete surfaces with constant negative Gaussian curvature and the Hirota equation
,” J. Diff. Geom.
43
, 527
–611
(1996
).11.
A.
Doliwa
and P. M.
Santini
, “Integrable dynamics of a discrete curve and the Ablowitz–Ladik hierarchy
,” J. Math. Phys.
36
, 1259
–1273
(1995
).12.
A.
Bobenko
and U.
Pinkall
, “Discrete isothermic surfaces
,” J. Reine Angew. Math.
475
, 187
–208
(1996
).13.
A. Bobenko and W. K. Schief, “Discrete affine spheres,” in Discrete Integrable Geometry and Physics, edited by A. Bobenko and R. Seiler (Oxford University Press, Oxford, 1999).
14.
R. Sauer, Differenzengeometrie (Springer-Verlag, Berlin, 1970).
15.
A.
Doliwa
and P. M.
Santini
, “Multidimensional quadrilateral lattices are integrable
,” Phys. Lett. A
233
, 365
–372
(1997
).16.
L. V.
Bogdanov
and B. G.
Konopelchenko
, “Lattice and q-difference Darboux–Zakharov–Manakov systems via method
,” J. Phys. A
28
, L173
–L178
(1995
).17.
R. R. Martin, J. de Pont, and T. J. Sharrock, “Cyclic surfaces in computer aided design,” in The Mathematics of Surfaces, edited by J. A. Gregory (Clarendon, Oxford, 1986), pp. 253–268.
18.
A. W. Nutbourne, “The solution of a frame matching equation,” in Ref. 17, pp. 233–252.
19.
A. Bobenko, “Discrete conformal maps and surfaces,” in Symmetries and Integrability of Difference Equations II, edited by P. Clarkson and F. Nijhoff (Cambridge University Press, Cambridge, 1999), pp. 97–108.
20.
J.
Cieśliński
, A.
Doliwa
, and P. M.
Santini
, “The integrable discrete analogues of orthogonal coordinate systems are multidimensional circular lattices
,” Phys. Lett. A
235
, 480
–488
(1997
).21.
A.
Doliwa
, S. V.
Manakov
, and P. M.
Santini
, “-reductions of the multidimensional quadrilateral lattice: the multidimensional circular lattice
,” Commun. Math. Phys.
196
, 1
–18
(1998
).22.
B. G.
Konopelchenko
and W. K.
Schief
, “Three-dimensional integrable lattices in Euclidean spaces: Conjugacy and orthogonality
,” Proc. R. Soc. London, Ser. A
454
, 3075
–3104
(1998
).23.
L. Bianchi, Lezioni di Geometria Differenziale (Zanichelli, Bologna, 1924).
24.
A.
Doliwa
, M.
Mañas
, L.
Martı́nez Alonso
, E.
Medina
, and P. M.
Santini
, “Multi-component KP hierarchy and classical transformations of conjugate nets
,” J. Phys. A
32
, 1197
–1216
(1999
).25.
A.
Doliwa
, “Geometric discretisation of the Toda system
,” Phys. Lett. A
234
, 187
–192
(1997
).26.
R.
Hirota
, “Discrete analogue of a generalized Toda equation
,” J. Phys. Soc. Jpn.
50
, 3785
–3791
(1981
).27.
M.
Mañas
, A.
Doliwa
, and P. M.
Santini
, “Darboux transformations for multidimensional quadrilateral lattices. I
,” Phys. Lett. A
232
, 99
–105
(1997
).28.
D.
Levi
and R.
Benguria
, “Bäcklund transformations and nonlinear differential-difference equations
,” Proc. Natl. Acad. Sci. USA
77
, 5025
–5027
(1980
).29.
D. E. Rowe, “The early geometrical works of Sophus Lie and Felix Klein,” in The History of Modern Mathematics, edited by D. E. Rowe and J. McCleary (Academic, New York, 1989), pp. 209–273.
30.
E. P. Lane, Projective Differential Geometry of Curves and Surfaces (University of Chicago Press, Chicago, 1932).
31.
L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces (Ginn and Company, Boston, 1909).
32.
A. Doliwa, “Discrete integrable geometry with ruler and compass,” in Ref. 19, pp. 122–136.
33.
E. V.
Ferapontov
, “Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences
,” J. Phys. A
30
, 6861
–6878
(1997
).34.
P. Samuel, Projective Geometry (Springer-Verlag, Berlin, 1988).
35.
L.
Lévy
, “Sur quelques équations linéares aux dérivées partieles du second ordre
,” J. Ecole Polytechnique
56
, 63
(1886
).36.
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer-Verlag, Berlin, 1991).
37.
W. Oevel and W. Schief, “Darboux theorems and the KP hierarchy,” in Application of Analytic and Geometric Methods to Nonlinear Differential Equations, edited by P. A. Clarkson (Kluwer, Dordrecht, 1992), pp. 193–206.
38.
B. G. Konopelchenko and W. K. Schief, “Lamé and Zakharov–Manakov systems: Combescure, Darboux and Bäcklund transformations,” Applied Mathematics preprint AM93/9, The University of New South Wales.
39.
H.
Jonas
, “Über die Transformation der konjugierten Systeme and über den gemeinsamen Ursprung der Bianchischen Permutabilitätstheoreme
,” Sitzungsber. Berl. Math. Ges.
14
, 96
–118
(1915
).40.
J. J. C.
Nimmo
and W. K.
Schief
, “Superposition principles associated with the Moutard transformation. An integrable discretisation of a -dimensional sine-Gordon system
,” Proc. R. Soc. London, Ser. A
453
, 255
–279
(1997
).41.
E. I.
Ganzha
and S. P.
Tsarev
, “On superposition of the auto-Bäcklund transformation for -dimensional integrable systems
,” solv-int/9606003.42.
M. J.
Ablowitz
, D.
Bar Yaacov
, and A. S.
Fokas
, “On the inverse scattering problem for the Kadomtsev–Petviashvili equation
,” Stud. Appl. Math.
69
, 135
(1983
).43.
L. V.
Bogdanov
and S. V.
Manakov
, “The nonlocal -problem and -dimensional soliton equations
,” J. Phys. A
21
, L537
–L544
(1988
).44.
L. V.
Bogdanov
and B. G.
Konopelchenko
, “Generalized integrable hierarchies and Combescure symmetry transformations
,” J. Phys. A
30
, 1591
–1603
(1997
).45.
L. V.
Bogdanov
and B. G.
Konopelchenko
, “Analytic-bilinear approach to integrable hierarchies II. Multicomponent KP and 2D Toda lattice hierarchies
,” J. Math. Phys.
39
, 4701
–4728
(1998
).46.
47.
R. Prus and A. Sym, “Rectilinear congruences and Bäcklund transformations: roots of the soliton theory,” in Proceedings of the 1st NOSONGE, Warsaw ’95 (Polish Scientific Publishers, Warsaw, 1998), pp. 25–36.
This content is only available via PDF.
© 2000 American Institute of Physics.
2000
American Institute of Physics
You do not currently have access to this content.
