On the formalism of local variational differential operators

The calculus of local variational differential operators introduced by B. L. Voronov, I. V. Tyutin, and Sh. S. Shakhverdiev is studied in the context of jet super space geometry. In a coordinate-free way, we relate these operators to variational multivectors, for which we introduce and compute the variational Poisson and Schouten brackets by means of a unifying algebraic scheme. We give a geometric definition of the algebra of multilocal functionals and prove that local variational differential operators are well defined on this algebra. To achieve this, we obtain some analytical results on the calculus of variations in smooth vector bundles, which may be of independent interest. In addition, our results give a new a new efficient method for finding Hamiltonian structures of differential equations

Coverings and the fundamental group for partial differential equations

Following I. S. Krasilshchik and A. M. Vinogradov, we regard systems of PDEs as manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and B\"acklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a discrete group, but a certain system of Lie groups. From this we deduce an algebraic necessary condition for two PDEs to be connected by a B\"acklund transformation. For the KdV equation and the nonsingular Krichever-Novikov equation these systems of Lie groups are determined by certain infinite-dimensional Lie algebras of Kac-Moody type. We prove that these two equations are not connected by any B\"acklund transformation. To achieve this, for a wide class of Lie algebras $\mathfrak{g}$ we prove that any subalgebra of $\mathfrak{g}$ of finite codimension contains an ideal of $\mathfrak{g}$ of finite codimension

On symmetries and cohomological invariants of equations possessing flat representations

We study the equation E_fc of flat connections in a fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1-1 correspondence with morphisms f: E\to E_fc, where E and E_fc are treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDE. In particular, the Lax pairs of the self-dual Yang-Mills equations and their reductions are of this type. With each flat representation we associate a complex C_f of vector-valued differential forms such that its first cohomology describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Backlund transformations. In addition, each higher infinitesimal symmetry S of E defines a 1-cocycle c_S of C_f. Symmetries with exact c_S form a subalgebra reflecting some geometric properties of E and f. We show that the complex corresponding to E_fc itself is 0-acyclic and 1-acyclic (independently of the bundle topology), which means that higher symmetries of E_fc are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries.Comment: 30 page

Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang--Baxter equation. Sergeev [S.M. Sergeev 1998 Lett. Math. Phys. 45, 113--119] presented classification results on three-dimensional tetrahedron maps obtained from the local Yang--Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang--Baxter equation for some simple $2\times 2$ matrix-functions, we show that there are (non-unique) solutions to the local Yang--Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang--Baxter equation, we obtain several new birational tetrahedron maps, including maps with matrix Lax representations on arbitrary groups, a $9$-dimensional map associated with a Darboux transformation for the derivative nonlinear Schr\"odinger (NLS) equation, and a $9$-dimensional generalisation of the $3$-dimensional Hirota map.Comment: 18 pages. New results added (section 4), and also the references list was update

Tetrahedron maps, Yang-Baxter maps, and partial linearisations

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang-Baxter maps, which are set-theoretical solutions to the quantum Yang-Baxter equation. In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Furthermore, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similar results on the differentials of Yang-Baxter and entwining Yang-Baxter maps are also presented. Using the obtained general results, we construct new examples of (parametric) Yang-Baxter and tetrahedron maps. The considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and M\"uller-Hoissen [arXiv:1708.05694] in a study of soliton solutions of vector Kadomtsev-Petviashvili (KP) equations. Also, we present invariants for this nonlinear tetrahedron map.Comment: 23 pages; v2: new results and references added, minor corrections mad

Miura-type transformations for lattice equations and Lie group actions associated with Darboux-Lax representations

Miura-type transformations (MTs) are an essential tool in the theory of integrable nonlinear partial differential and difference equations. We present a geometric method to construct MTs for differential-difference (lattice) equations from Darboux–Lax representations (DLRs) of such equations. The method is applicable to parameter-dependent DLRs satisfying certain conditions. We construct MTs and modified lattice equations from invariants of some Lie group actions on manifolds associated with such DLRs. Using this construction, from a given suitable DLR one can obtain many MTs of different orders. The main idea behind this method is closely related to the results of Drinfeld and Sokolov on MTs for the partial differential KdV equation. Considered examples include the Volterra, Narita–Itoh–Bogoyavlensky, Toda, and Adler–Postnikov lattices. Some of the constructed MTs and modified lattice equations seem to be new

Conservation laws for multidimensional systems and related linear algebra problems

We consider multidimensional systems of PDEs of generalized evolution form with t-derivatives of arbitrary order on the left-hand side and with the right-hand side dependent on lower order t-derivatives and arbitrary space derivatives. For such systems we find an explicit necessary condition for existence of higher conservation laws in terms of the system's symbol. For systems that violate this condition we give an effective upper bound on the order of conservation laws. Using this result, we completely describe conservation laws for viscous transonic equations, for the Brusselator model, and the Belousov-Zhabotinskii system. To achieve this, we solve over an arbitrary field the matrix equations SA=A^tS and SA=-A^tS for a quadratic matrix A and its transpose A^t, which may be of independent interest.Comment: 12 pages; proof of Theorem 1 clarified; misprints correcte

Formation of university students safety culture in modern socio-cultural and technogenic conditions

© 2017 Serials Publications.The urgency of the paper is determined by the sharp growth of personal, social, physical threats to the security of the individual in the modern world under the influence of social tension and technological factors. The purpose of the paper is the development and approbation of the psychological and pedagogical model for the formation of university students' safety culture.In the framework of the study of the educational environment, the authors identify the problem of high risks to personal security and the need to form and develop students' safe culture.The authors suggest a new interpretation of the individual's security culture, its essence and components, corresponding to contemporary socio-cultural and technological conditions.The authors have defined the methods and mechanisms, the diagnostic tools for assessing the formation of university students' safety culture, and the results of the experiment.The paper is intended for educators, researchers and specialists in the field of vocational education and safety