On the variational interpretation of the discrete KP equation

We study the variational structure of the discrete Kadomtsev-Petviashvili (dKP) equation by means of its pluri-Lagrangian formulation. We consider the dKP equation and its variational formulation on the cubic lattice \(\mathbb Z^{N}\) as well as on the root lattice \(Q(A_{N})\). We prove that, on a lattice of dimension at least four, the corresponding Euler-Lagrange equations are equivalent to the dKP equation.