Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrödinger operator

Nieszporski, Maciej; Santini, P. M.; Doliwa, Adam

doi:10.1016/j.physleta.2004.02.003

Cited by 10 publications

(52 citation statements)

References 22 publications

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“…As it was shown in [24], equation (2) is an integrable discretization of the elliptic (if AB > 1) equation…”

Section: Introductionmentioning

confidence: 91%

“…then the function Ψ, restricted to the even grid Z 2 e , satisfies the discrete Schrödinger equation [24] …”

Section: Different Gauge Forms Of the 5-point Schemementioning

confidence: 99%

“…Here we show, for instance, the construction of the Darboux transformations [24] of the 5-point scheme in the affine (11), equal-field (37) and Schrödinger (41) forms, induced by the transformations of the 4-point lattice (20).…”

Section: Darboux Transformations Of the Sub-latticementioning

confidence: 99%

“…The resulting transformation formulas differ from (58)-(59) in some shifts. The present form has been chosen to be in agreement with the earlier formulas of [24].…”

Section: Dts Of the Affine 5-point Schemementioning

confidence: 99%

“…The 5-point scheme (2), introduced in [24] as an integrable discretization of equation (4), describes the normal vector of a quadrilateral lattice in R 3 , and this embedding is obtained via a suitable generalization of the Lelieuvre formulas [25]. It is remarkable that equations (1) and (2), which are basic equations in the recently developed theory of the Integrable Discrete Geometries, are also the basic equations of an integrable discretization of the theory of holomorphic and harmonic functions.…”

Section: Introductionmentioning

confidence: 99%

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Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

Doliwa

Grinevich

Nieszporski

et al. 2007

Journal of Mathematical Physics

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We introduce the sub-lattice approach, a procedure to generate, from a given integrable lattice, a sub-lattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sub-lattice, 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). We use the sub-lattice 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. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in R 3 .

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“…As it was shown in [24], equation (2) is an integrable discretization of the elliptic (if AB > 1) equation…”

Section: Introductionmentioning

confidence: 91%

“…then the function Ψ, restricted to the even grid Z 2 e , satisfies the discrete Schrödinger equation [24] …”

Section: Different Gauge Forms Of the 5-point Schemementioning

confidence: 99%

Section: Darboux Transformations Of the Sub-latticementioning

confidence: 99%

“…The resulting transformation formulas differ from (58)-(59) in some shifts. The present form has been chosen to be in agreement with the earlier formulas of [24].…”

Section: Dts Of the Affine 5-point Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

Doliwa

Grinevich

Nieszporski

et al. 2007

Journal of Mathematical Physics

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The Self-Adjoint 5-Point and 7-Point Difference Operators, the Associated Dirichlet Problems, Darboux Transformations and Lelieuvre Formulae

Nieszporski¹,

Santini²

2005

Glasgow Math. J.

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Abstract. We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow us to solve Dirichlet boundary value problems; then we present their Moutard transformations (distinguished examples of transformation of Darboux type in two dimensions). Finally we construct their Lelieuvre formulae and we show that, at the level of the normal vector and in full analogy with their continuous counterparts, the self-adjoint 5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar), while the self-adjoint 7-point scheme characterizes a generic 2D lattice.2000 Mathematics Subject Classification. 35J55. Introduction.In the last two decades of the 19th century and in the beginning of the 20th century many great mathematicians (Bianchi, Darboux and others) developed a differential geometry studying transformations of certain geometric structures and proved theorems of permutability of such transformations, obtaining in turn nonlinear superposition principles for the nonlinear differential equations characterizing the above geometries. These results can be viewed as the pre-history [1, 2] of the modern theory of integrable nonlinear systems, which is based on the existence of linear differential operators possessing symmetry transformations of Darboux type (the Darboux Transformations (DTs)) and nonlinear isospectral symmetries (the integrable nonlinear systems).After more than one century, in studying discrete integrable systems (which are, in a sense, richer and more fundamental than their continuous counterparts and, for these reasons, worth studying), one often makes use of the mutual interplay between geometry and the theory of integrable systems in the discrete case too [3].The main goal of this paper is to present two examples of such an interplay; we start with the self-adjoint 7-point operator

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Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

Doliwa

Nieszporski

Santini

2007

Journal of Mathematical Physics

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Abstract. An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete 3D system is constructed.

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Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrödinger operator

Cited by 10 publications

References 22 publications

Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

The Self-Adjoint 5-Point and 7-Point Difference Operators, the Associated Dirichlet Problems, Darboux Transformations and Lelieuvre Formulae

Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

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