## Vol 63, No 3 (2017): Diﬀerential and Functional Diﬀerential Equations

Articles
On Lacunas in the Lower Part of the Spectrum of the Periodic Magnetic Operator in a Strip
Borisov D.I.
###### Abstract
We consider the Schro¨dinger periodic magnetic operator in an inﬁnite ﬂat straight strip. We prove that if the magnetic potential satisﬁes certain conditions and the period is small enough, then the lower part of the band spectrum has no inner lacunas. The length of the lower part of the band spectrum with no inner lacunas is calculated explicitly. The upper estimate for the small parameter allowing these results is calculated as a number as well.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):373-391  Optimal Perturbations of Systems with Delayed Argument for Control of Dynamics of Infectious Diseases Based on Multicomponent Actions
Bocharov G.A., Nechepurenko Y.M., Khristichenko M.Y., Grebennikov D.S.
###### Abstract
In this paper, we apply optimal perturbations to control mathematical models of infectious diseases expressed as systems of nonlinear diﬀerential equations with delayed argument. We develop the method for calculation of perturbations of the initial state of a dynamical system with delayed argument producing maximal ampliﬁcation in the given local norm taking into account weights of perturbation components. For the model of experimental virus infection, we construct optimal perturbation for two types of stationary states, with low or high virus load, corresponding to diﬀerent variants of chronic virus infection ﬂow.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):392-417  Lagrangian Representations for Linear and Nonlinear Transport
Bianchini S., Bonicatto P., Marconi E.
###### Abstract
In this note we present a unifying approach for two classes of ﬁrst order partial diﬀerential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector ﬁeld. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):418-436  Method of Monotone Solutions for Reaction-Diﬀusion Equations
Volpert V., Vougalter V.
###### Abstract
Existence of solutions of reaction-diﬀusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reactiondiﬀusion systems for which there exist two subclasses of solutions separated in the function space, monotone and non-monotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):437-454  Dynamical Systems and Topology of Magnetic Fields in Conducting Medium
Grines V.Z., Zhuzhoma E.V., Pochinka O.V.
###### Abstract
We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic ﬁelds in conducting media. For substantial classes of magnetic ﬁelds, we consider well-known physical models allowing us to reduce investigation of such ﬁelds to study of vector ﬁelds and Morse-Smale diﬀeomorphisms as well as diﬀeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point-charge magnetic ﬁeld model, we consider the problem of separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic ﬁelds in the solar corona and solve the problem of topological equivalency of ﬁelds in this class. We develop a topological modiﬁcation of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diﬀeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):455-474  On Entropy Solutions of Anisotropic Elliptic Equations with Variable Nonlinearity Indices
Kozhevnikova L.M.
###### Abstract
For a certain class of second-order anisotropic elliptic equations with variable nonlinearity indices and L1 right-hand side we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable indices.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):475-493  On the Volume Formula for Hyperbolic 4-Dimensional Simplex
Krasnov V.A.
###### Abstract
In this paper, we derive an explicit formula for the volume of abritrary hyperbolic 4-simplex depending on vertices coordinates.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):494-503  On Ellipticity of Hyperelastic Models Based on Experimental Data
Salamatova V.Y., Vasilevskii Y.V.
###### Abstract
The condition of ellipticity of the equilibrium equation plays an important role for correct description of mechanical behavior of materials and is a necessary condition for new deﬁning relationships. Earlier, new deformation measures were proposed to vanish correlations between the terms, that dramatically simpliﬁes restoration of deﬁning relationships from experimental data. One of these new deformation measures is based on the QR-expansion of deformation gradient. In this paper, we study the strong ellipticity condition for hyperelastic material using the QR-expansion of deformation gradient.
Contemporary Mathematics. Fundamental Directions. 2017;63(3):504-515  Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2
Sevryuk M.B.
###### Abstract
We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the ﬁxed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coeﬃcient matrix of the variational equation along the torus).
Contemporary Mathematics. Fundamental Directions. 2017;63(3):516-541  