A Laplace ladder of discrete Laplace equations

The notion of a Laplace ladder for a discrete analogue of the Laplace equation is presented. The adjoint of the discrete Moutard equation and a discrete counterpart of the nonlinear form of Goursat equation are introduced. 1 Notation Value of functions of continuous variables we denote by f(u, v) i.e. f : R ∋ (u, v) 7→ f(u, v) ∈ R while functions of discrete variables by f(m1, m2) i.e. f : Z 2 ∋ (m1, m2) 7→ f(m1, m2) ∈ R. Partial derivatives are denoted by comma e.g. f,uv (u, v) := ∂2f ∂u∂v (u, v). Shift operators are denoted by subscripts in brackets e.g. f(m1, m2)(1) := f(m1 + 1, m2), f(m1, m2)(2) := f(m1, m2 + 1), f(m1, m2)(12) := f(m1 + 1, m2 + 1), f(m1, m2)(−1) := f(m1 − 1, m2) etc. We omit arguments when operators indicate what kind of functions (of discrete or continuous variables) we deal with. Difference operators are denoted by ∆if := f(i) − f . The diamond operator we define as follows 3f := f(12)f f(1)f(2) .