An integrable multicomponent quad-equation and its Lagrangian formulation

We present a hierarchy of discrete systems whose first members are the lattice modified Korteweg–de Vries equation and the lattice modified Boussinesq equation. The $N$th member in the hierarchy is an $N$-component system defined on an elementary plaquette in the two-dimensional lattice. The system is multidimensionally consistent, and we obtain a Lagrangian that respects this feature, i.e., has the desirable closure property.