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

Marcella Palese, Fabrizio Zanello

TL;DR

The work provides a geometric variational framework for $2D$ Boussinesq-type models by formulating a two-field Lagrangian system on higher-order jet spaces and constructing full Lepage equivalents of Krupka–Betounes type. Through geometric integration by parts and recursive Lepage constructions, it derives new $2D$ fourth- and sixth-order Boussinesq-type equations with mixed spatial derivatives, and obtains a two-field variational characterization of the stationary reduction of the moving-frame KP equation. The approach links EL dynamics, conservation laws, and higher-order variational structures, offering a route to refined conserved currents and symmetry analysis within a unified Lagrangian-geometric setting. These results extend the applicability of Lepage-equivalent methods to complex $(2+1)$-dimensional dispersive systems and highlight the role of extended variational content in integrable and near-integrable models.

Abstract

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.

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

TL;DR

The work provides a geometric variational framework for Boussinesq-type models by formulating a two-field Lagrangian system on higher-order jet spaces and constructing full Lepage equivalents of Krupka–Betounes type. Through geometric integration by parts and recursive Lepage constructions, it derives new fourth- and sixth-order Boussinesq-type equations with mixed spatial derivatives, and obtains a two-field variational characterization of the stationary reduction of the moving-frame KP equation. The approach links EL dynamics, conservation laws, and higher-order variational structures, offering a route to refined conserved currents and symmetry analysis within a unified Lagrangian-geometric setting. These results extend the applicability of Lepage-equivalent methods to complex -dimensional dispersive systems and highlight the role of extended variational content in integrable and near-integrable models.

Abstract

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

Paper Structure

This paper contains 12 sections, 3 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Framework: geometry and symmetry
  3. Extension of the Poincaré--Cartan form to a 'full' equivalent
  4. Geometric integration by parts for $k$-contact forms of lower horizontal degree
  5. Lepage equivalents
  6. Recursive formulae and 'full' Lepage equivalents
  7. Applications: two-fields Lagrangians generating $2D$ modified and extended Boussinesq equations
  8. Fourth-order $(2+1)$ extensions with mixed spatial derivatives
  9. Two-field derivation of fourth-order $(2+1)$ extensions
  10. Sixth-order $(2+1)$ extensions with mixed spatial derivatives
  11. The case of the reduced KP equation in the moving-frame
  12. Symmetries and 'improved' conserved currents

Key Result

Proposition 3.4

Let $\lambda = \mathcal{L} ds$ be a Lagrangian on $J^r Y$.

Theorems & Definitions (13)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Theorem 5.1
  • ...and 3 more