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

Abstract

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 $\RR^3$.

arXiv comment31 pages, 14 figures

References in corpus (4)

• Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrodinger operator
• Discrete analogs of the Darboux-Egoroff metrics
• Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function
• The self-adjoint 5-point and 7-point difference operators, the associated Dirichlet problems, Darboux transformations and Lelieuvre formulas

Cited by in corpus (3)

• Discrete differential geometry. Consistency as integrability
• The B-quadrilateral lattice, its transformations and the algebro-geometric construction
• Integrable Systems and Discrete Geometry