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

In the frame of the Lagrangian formalism on [Formula: see text]-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 [Formula: see text] modified Boussinesq equation, 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 [Formula: see text]-contact component of their exterior differential. As a main result, we find new 2D fourth- and sixth-order modified Boussinesq-type equations, containing mixed terms in both the spatial variables [Formula: see text] and [Formula: see text]. As a byproduct, we also obtain a 2-field variational characterization of the stationary reduction of the moving-frame (according to Bogdanov and Zakharov) KP equation.