Lepage equivalents for second order Lagrangians and applications: $2D$ modified higher order Boussinesq-type equations
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arXiv
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| Natura: | Preprint |
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2025
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| author | Palese, Marcella Zanello, Fabrizio |
| author_facet | Palese, Marcella Zanello, Fabrizio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lepage equivalents for second order Lagrangians and applications: $2D$ modified higher order Boussinesq-type equations Palese, Marcella Zanello, Fabrizio Mathematical Physics 58A20, 37K06, 37K25, 37K58, 53Zxx, 81R12 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. |
| title | Lepage equivalents for second order Lagrangians and applications: $2D$ modified higher order Boussinesq-type equations |
| topic | Mathematical Physics 58A20, 37K06, 37K25, 37K58, 53Zxx, 81R12 |
| url | https://arxiv.org/abs/2510.07090 |