Lepage equivalents for second order Lagrangians and applications: $2D$ modified higher order Boussinesq-type equations

In the frame of the Lagrangian formalism on $r$-order prolongations of fibered manifolds and related structures such as (prolongation of) projectable vector fields, (sheaves of) differential forms and contact structures, we propose a Lagrangian two-field derivation of $2D$ modified Boussinesq equations, obtained as coupled systems of Euler--Lagrange (E-L) equations for the two fields. By means of a recursive formula involving geometric integration by parts formulae, we construct extended `full' equivalents of such Lagrangians, in particular of Krupka--Betounes type, by which the equations are obtained straightly as the $1$-contact component of their exterior differential. As a main result we find {\em new $2D$ fourth- and sixth-order modified Boussinesq-type equations}, containing mixed terms in both the spatial variables $x$ and $y$. As a byproduct, we also obtain a {\em $2$-field variational characterization} of the stationary reduction of the moving-frame (according to Bogdanov and Zakharov) KP equation.