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 con struct 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 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 2-field variational characterization of the stationary reduction of the moving-frame (according to Bogdanov and Zakharov) KP equation.
3. Renormalization of Higher Currents of the sine-Gordon Model in pAQFT.
Abstract:In this paper we show that the higher currents of the sine-Gordon model are super-renormalizable by power counting in the framework of pAQFT. First we obtain closed recursive formulas for the higher currents in the classical theory and introduce a suitable notion of degree for their components. We then move to the pAQFT setting and, by means of some technical results, we compute explicit formulas for the unrenormalized interacting currents. Finally, we perform what we call the piecewise renormalization of the interacting higher currents, showing that the renormalization process involves a number of steps which is bounded by the degree of the classical conserved currents.
2.Second order Lagrangians for (2 + 1)-dimensional generalized Boussinesq equations and an extension of the Krupka-Betounes equivalent.
Abstract:We determine second order Lagrangians for (2 + 1)-dimensional generalized Boussinesq equations and we discuss some aspects concerning conservation laws associated with invariance properties of their extended ‘full’ equivalents, in particular of Krupka–Betounes type. Such equivalents are constructed by means of a recursive formula involving geometric integration by parts formulae.
1.Geometric integration by parts and Lepage equivalents.
Abstract:We compare theintegration by partsof contact forms – leading to the definition of the interior Euler operator – with the so-called canonical splittings of variational morphisms. In particular, we discuss the possibility of a generalization of the first method to contact forms of lower degree. We define a suitable Residual operator for this case and, working out an original conjecture by Olga Rossi, we recover the Krupka–Betounes equivalent for first order field theories. A generalization to the second order case is discussed.