Integrable lattices and their sublattices: 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 R 3 .

The B-quadrilateral lattice, its transformations and the algebro-geometric construction

“…Such curves have been already used in construction of solutions of the BKP hierarchy [14] and of its two-component generalization [47]. We develop the corresponding results of [22] and we present the explicit formulas for the lattice points and the solutions of the discrete BKP equation in terms of the Prym theta functions related to such special curves.…”

“…For an arbitrary m ∈ Z N there exists [22] the unique function ψ(m) meromorphic on Γ having in points Q i (in points σ(Q i )) poles (corrspondingly, zeros) of the order m i , no other singularities except for possible simple poles in points of the divisor D, and normalized to 1 at Q ∞ . In [22] it was shown that, as a function of the discrete parameter m, the wave function ψ satisies the system of the discrete Moutard equations…”

“…Two dimensional quadrilateral lattices whose homogeneous coordinates satisfy (up to a gauge) equation (2.9) are characterized geometrically [22] by condition that any point x and its four second-order neighbours x (±1±2) are contained in a subspace of dimension three. Obviously, any two dimensional slide of the B-quadrilateral lattice fulfills this property, which can therefore serve as definition of a two dimensional BQL.…”

“…It is known [22] that in the BQL reduction case, the generic algebraic curve, used to generate solutions of the discrete Darboux equations [1], should be replaced by a curve admitting a holomorphic involution with two fixed points. Such curves have been already used in construction of solutions of the BKP hierarchy [14] and of its two-component generalization [47].…”

“…In doing that we start from recent results of [22], where restrictions on the algebro-geometric data of the discrete Darboux system [1] compatible with the discussed reduction were given. Then we proceed to formulas for the wave and τ -functions of the BQL in terms of the Prym theta functions related with the algebraic curves used in the construction.…”