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

New Results

On Lacunas in the Lower Part of the Spectrum of the Periodic Magnetic Operator in a Strip

Borisov D.I.


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
pages 373-391 views

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.


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
pages 392-417 views

Lagrangian Representations for Linear and Nonlinear Transport

Bianchini S., Bonicatto P., Marconi E.


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
pages 418-436 views

Method of Monotone Solutions for Reaction-Diffusion Equations

Volpert V., Vougalter V.


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
pages 437-454 views

Dynamical Systems and Topology of Magnetic Fields in Conducting Medium

Grines V.Z., Zhuzhoma E.V., Pochinka O.V.


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
pages 455-474 views

On Entropy Solutions of Anisotropic Elliptic Equations with Variable Nonlinearity Indices

Kozhevnikova L.M.


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
pages 475-493 views

On the Volume Formula for Hyperbolic 4-Dimensional Simplex

Krasnov V.A.


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
pages 494-503 views

On Ellipticity of Hyperelastic Models Based on Experimental Data

Salamatova V.Y., Vasilevskii Y.V.


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
pages 504-515 views

Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

Sevryuk M.B.


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
pages 516-541 views

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