General classification of integrable two-dimensional lattices of the form un,xy = f(un+1, un, un−1, un,x, un,y)

Classify all nonlinear two-dimensional lattice equations of the form un,xy = f(un+1, un, un−1, un,x, un,y) (equation (1.3)) for which there exist boundary functions f1 and f2 such that, for every integer m ≥ 2, the finite-field reductions u1,xy = f1(u1, u2, u1,x, u1,y), uj,xy = f(u_{j+1}, uj, u_{j−1}, uj,x, uj,y) for 1 < j < m, and um,xy = f2(um, um−1, um,x, um,y) are integrable in the sense of Darboux (i.e., admit complete sets of characteristic integrals in both characteristic directions), without assuming the quasilinear structure (1.5) for f.

Background

The paper proposes a classification approach for 3D lattices based on the existence of a hierarchy of finite-field reductions that are Darboux integrable, meaning each reduced hyperbolic system admits complete sets of characteristic integrals in both characteristic directions. This approach was applied to equations of the general form un,xy = f(un+1, un, un−1, un,x, un,y) (equation (1.3)).

Within the quasilinear subclass (1.5), where the dependence on first derivatives is linear, the classification problem was completely solved, yielding the list (E1)–(E7). However, extending this classification beyond the quasilinear assumption to the full generality of (1.3) remains unresolved.