We present the sublattice approach, a procedure to generate, from a given integrable lattice, a sublattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sublattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable discrete geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). Therefore an integrable, at one energy, discretization of elliptic two-dimensional operators is considered. We use the sublattice point of view to derive, from the Darboux transformations and superposition formulas of the discrete Moutard equation, the Darboux transformations and superposition formulas of the self-adjoint 5-point scheme. We also construct, from algebro-geometric solutions of the discrete Moutard equation, algebro-geometric solutions of the self-adjoint 5-point scheme. In particular, we show that the corresponding restrictions on the finite-gap data are of the same type as those for the fixed energy problem for the two-dimensional Schrödinger operator. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in .
REFERENCES
1.
Adler
, V. E.
and Suris
, Y. B.
, “: Integrable master equation related to an elliptic curve
,” Int. Math. Res. Notices
47
, 2523
–2553
(2004
).2.
Adler
, V. E.
, Bobenko
, A. I.
, and Suris
, Y. B.
, “Classification of integrable equations on quad-graphs. The consistency approach
,” Commun. Math. Phys.
233
, 513
–543
(2003
).3.
Akhmetshin
, A. A.
, Krichever
, I. M.
, and Volvovski
, Y. S.
, “Discrete analogs of the Darboux-Egoroff metrics
,” Proc. Steklov Inst. Math.
225
, 16
–39
(1999
).4.
Andrews
, G. E.
, Baxter
, R. J.
, and Forrester
, P. J.
, “Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
,” J. Stat. Phys.
35
, 193
–266
(1984
).5.
6.
Bobenko
, A.
and Pinkall
, U.
, “Discrete surfaces with constant negative Gaussian curvature and the Hirota equation
,” J. Diff. Geom.
43
, 527
–611
(1996
).7.
Bobenko
, A. I.
and Suris
, Y. B.
, “Integrable systems on quad-graphs
,” Int. Math. Res. Notices
11
, 573
–611
(2002
).8.
Bobenko
, A. I.
, Mercat
, Ch.
, and Suris
, Y. B.
, “Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function
,” J. Reine Angew. Math.
583
, 117
–161
(2005
).9.
Konopelchenko
, B. G.
and Pinkall
, U.
, “Projective generalizations of Lelieuvre’s formula
,” Geom. Dedic.
79
, 81
–99
(2000
).10.
Cherednik
, I. V.
, “Reality conditions in “finite-zone” integration
,” Dokl. Akad. Nauk SSSR
252
, 1104
–1108
(1980
).11.
Date
, E.
, Jimbo
, M.
, and Miwa
, T.
, “Method for generating discrete soliton equations. V
,” J. Phys. Soc. Jpn.
52
, 766
–771
(1983
).12.
Date
, E.
, Jimbo
, M.
, Kuniba
, A.
, Miwa
, T.
, and Okado
, M.
, “Exactly solvable SOS models. Local height probabilities and theta function identities
,” Nucl. Phys. B
290
, 231
–273
(1987
).13.
Doliwa
, A.
and Santini
, P. M.
, “The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
,” J. Geom. Phys.
36
, 60
–102
(2000
).14.
Dubrovin
, B. A.
, Krichever
, I. M.
, and Novikov
, S. P.
, “Schrödinger equation in magnetic field and Riemann surfaces
,” Sov. Math. Dokl.
17
, 947
–951
(1976
).15.
Duffin
, R. J.
, “Basic properties of discrete analytic functions
,” Duke Math. J.
23
, 335
–363
(1956
).16.
Duffin
, R. J.
, “Potential theory on a rhombic lattice
,” J. Comb. Theory, Ser. A
5
, 258
–272
(1968
).17.
Dynnikov
, I. A.
and Novikov
, S. P.
, “Geometry of the triangle equation on two-manifolds
,” Mosc. Math. J.
3
, 419
–438
(2003
).18.
Fay
, J. D.
, Theta Functions on Riemann Surfaces
(Springer-Verlag
, Berlin
, 1973
).19.
Ferrand
, J.
, “Fonctions preharmoniques et fonctions preholomorphes
,” Bull. Sci. Math.
68
, 152
–180
(1944
).20.
Grinevich
, P. G.
and Novikov
, S. P.
, “Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions. 1. Energies below the ground state
,” Funct. Anal. Appl.
22
, 19
–27
(1988
).21.
Kac
, V. G.
and van de Leur
, J.
, in The n-component KP Hierarchy and Representation Theory
, Important Developments in Soliton Theory
, edited by A. S.
Fokas
, and V. E.
Zakharov
,, (Springer
, Berlin
, 1993
), pp. 302
–343
.22.
Kenyon
, R.
, “The Laplacian and Dirac operators on critical planar graphs
,” Invent. Math.
150
, 409
–439
(2002
).23.
Krichever
, I. M.
, “An algebraic-geometric construction of the Zakharov-Shabat equations and their periodic solutions
,” Sov. Math. Dokl.
17
, 394
–397
(1976
).24.
Krichever
, I. M.
, “Two-dimensional periodic difference operators and algebraic geometry
,” Dokl. Akad. Nauk SSSR
285
, 31
–36
(1985
).25.
Kulish
, P. P.
, Reshetikhin
, N. Yu.
, and Sklyanin
, E. K.
, “Yang-Baxter equation and representation theory: I
,” Lett. Math. Phys.
5
, 393
–403
(1981
).26.
Levi
, D.
and Benguria
, R.
, “Bäcklund transformations and nonlinear differential-difference equations
,” Proc. Natl. Acad. Sci. U.S.A.
77
, 5025
–5027
(1980
).27.
Manakov
, S. V.
, “The method of the inverse scattering problem, and two-dimensional evolution equations
,” Usp. Mat. Nauk
31
, 245
–246
(1976
) (in Russian).28.
Mercat
, Ch.
, “Discrete Riemann surfaces and the Ising model
,” Commun. Math. Phys.
218
, 177
–216
(2001
);Ph.D. thesis,
Université Louis Pasteur
(Strasbourg) 1998
, http://www-irma.u-strasbg.fr/irma/publications/1998/98014.shtml29.
Mercat
, Ch.
, “Exponentials form a basis of discrete holomorphic functions on a compact
,” Bull. Soc. Math. France
132
, 305
–326
(2004
).30.
Miwa
, T.
, “On Hirota’s difference equations
,” Proc. Jpn. Acad., Ser. A: Math. Sci.
58
, 9
–12
(1982
).31.
Mumford
, D.
, Tata Lectures on Theta I
(Birkhäuser Boston
, Boston, MA
, 1983
).32.
Natanzon
, S. M.
, “Real nonsingular finite zone solutions of soliton equations
,” Am. Math. Soc. Transl.
170
, 153
–183
(1995
).33.
Nieszporski
, M.
and Santini
, P. M.
, “The self-adjoint 5-point and 7-point difference operators, the associated Dirichlet problems, Darboux transformations and Lelieuvre’s formulas
,” Glasg. Math. J.
47A
, 133
–147
(2005
).34.
Nieszporski
, M.
, Santini
, P. M.
, and Doliwa
, A.
, “Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrödinger operator
,” Phys. Lett. A
323
, 241
–250
(2004
).35.
Nijhoff
, F. W.
and Capel
, H. W.
, “The discrete Korteweg-de Vries equation
,” Acta Appl. Math.
39
, 133
–158
(1995
).36.
Nijhoff
, F. W.
, Quispel
, G. R. W.
, and Capel
, H. W.
, “Direct linearization of nonlinear difference difference equations
,” Phys. Lett.
97A
, 125
–128
(1983
).37.
Nimmo
, J. J. C.
and Schief
, W. K.
, “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
).38.
Novikov
, S. P.
, “A periodic problem for the Korteweg-de Vries equation. I
,” Funkc. Anal. Priloz.
8
, 54
–66
(1974
) (in Russian).39.
Novikov
, S. P.
and Dynnikov
, I. A.
, “Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds
,” Russ. Math. Surveys
52
, 1057
–1116
(1997
).40.
Novikov
, S. P.
and Veselov
, A. P.
, “Finite-zone, two-dimensional Schrödinger operators. Potential operators
,” Sov. Math. Dokl.
30
, 705
–708
(1984
);Novikov
, S. P.
and Veselov
, A. P.
,“Finite-zone, two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations
,” Sov. Math. Dokl.
30
, 588
–591
(1984
).41.
Ocneanu
, A.
, in Operator Algebras and Applications
, London Mathematical Society Lecture Note Series Vol. 136
(Cambridge University Press
, Cambridge
, 1988
), Vol. 2
, pp. 119
–172
.42.
Pasquier
, V.
, “Two-dimensional critical systems labelled by Dynkin diagrams
,” Nucl. Phys. B
285
, 162
–172
(1987
).43.
Roche
, Ph.
, “Ocneanu cell calculus and integrable lattice models
,” Commun. Math. Phys.
127
, 395
–424
(1990
).44.
45.
Thurston
, W. P.
, “The finite Riemann mapping theorem
,” Invited talk at the International Symposium on the Occasion of the Proof of the Bieberbach Conjecture
, Purdue University, 1985
(unpublished).46.
Ward
, R. S.
, “Integrable and solvable systems, and relations among them
,” Philos. Trans. R. Soc. London, Ser. A
315
, 451
–457
(1985
).© 2007 American Institute of Physics.
2007
American Institute of Physics
You do not currently have access to this content.
