Integrable lattices and their sublattices: From the discrete Moutard (discrete CauchyRiemann) 4point equation to the selfadjoint 5point scheme
Abstract
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 4point equation and its sublattice, the selfadjoint 5point 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 CauchyRiemann equation). Therefore an integrable, at one energy, discretization of elliptic twodimensional 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 selfadjoint 5point scheme. We also construct, from algebrogeometric solutions of the discrete Moutard equation, algebrogeometric solutions of the selfadjoint 5point scheme. In particular, we show that the corresponding restrictions on the finitegap data are of the same type as those for the fixed energy problem for the twodimensional Schroedinger operator. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in R{sup 3}.
 Authors:
 Wydzial Matematyki i Informatyki, Uniwersytet WarminskoMazurski w Olsztynie, ulica Zolnierska 14 A, 10561 Olsztyn (Poland)
 (Russian Federation)
 (Poland) and School of Mathematics, University of Leeds, LS2 9JT Leeds (United Kingdom)
 (Italy) and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Piazzale Aldo Moro 2, I00185 Rome (Italy)
 Publication Date:
 OSTI Identifier:
 20929624
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2406056; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; FUNCTIONS; GEOMETRY; INTEGRAL CALCULUS; INTEGRAL EQUATIONS; MATHEMATICAL OPERATORS; MATHEMATICAL SOLUTIONS; SCHROEDINGER EQUATION; TETRAGONAL LATTICES; TRANSFORMATIONS; TWODIMENSIONAL CALCULATIONS
Citation Formats
Doliwa, A., Grinevich, P., Nieszporski, M., Santini, P. M., Landau Institute for Theoretical Physics, 117940 Moscow, Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ulica Hoza 74, 00682 Warsaw, and Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I00185 Rome. Integrable lattices and their sublattices: From the discrete Moutard (discrete CauchyRiemann) 4point equation to the selfadjoint 5point scheme. United States: N. p., 2007.
Web. doi:10.1063/1.2406056.
Doliwa, A., Grinevich, P., Nieszporski, M., Santini, P. M., Landau Institute for Theoretical Physics, 117940 Moscow, Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ulica Hoza 74, 00682 Warsaw, & Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I00185 Rome. Integrable lattices and their sublattices: From the discrete Moutard (discrete CauchyRiemann) 4point equation to the selfadjoint 5point scheme. United States. doi:10.1063/1.2406056.
Doliwa, A., Grinevich, P., Nieszporski, M., Santini, P. M., Landau Institute for Theoretical Physics, 117940 Moscow, Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ulica Hoza 74, 00682 Warsaw, and Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I00185 Rome. Mon .
"Integrable lattices and their sublattices: From the discrete Moutard (discrete CauchyRiemann) 4point equation to the selfadjoint 5point scheme". United States.
doi:10.1063/1.2406056.
@article{osti_20929624,
title = {Integrable lattices and their sublattices: From the discrete Moutard (discrete CauchyRiemann) 4point equation to the selfadjoint 5point scheme},
author = {Doliwa, A. and Grinevich, P. and Nieszporski, M. and Santini, P. M. and Landau Institute for Theoretical Physics, 117940 Moscow and Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ulica Hoza 74, 00682 Warsaw and Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I00185 Rome},
abstractNote = {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 4point equation and its sublattice, the selfadjoint 5point 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 CauchyRiemann equation). Therefore an integrable, at one energy, discretization of elliptic twodimensional 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 selfadjoint 5point scheme. We also construct, from algebrogeometric solutions of the discrete Moutard equation, algebrogeometric solutions of the selfadjoint 5point scheme. In particular, we show that the corresponding restrictions on the finitegap data are of the same type as those for the fixed energy problem for the twodimensional Schroedinger operator. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in R{sup 3}.},
doi = {10.1063/1.2406056},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 48,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}

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