# Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point 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 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 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 Warminsko-Mazurski w Olsztynie, ulica Zolnierska 14 A, 10-561 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, I-00185 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; TWO-DIMENSIONAL 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, 00-682 Warsaw, and Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I-00185 Rome.
```*Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point 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, 00-682 Warsaw, & Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I-00185 Rome.
```*Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point 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, 00-682 Warsaw, and Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I-00185 Rome. Mon .
"Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme". United States.
doi:10.1063/1.2406056.
```

```
@article{osti_20929624,
```

title = {Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point 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, 00-682 Warsaw and Dipartimento di Fisica, Universita di Roma 'La Sapienza', Piazzale Aldo Moro 2, I-00185 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 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 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}

}