An effective model of a porous-fluid medium

For a medium containing alternating porous Biot layers and fluid layers, an effective model is established by the method of matrix averaging. An investigation of equations of this effective model shows that the wave field consists of a leading front and two triangular fronts. The velocities of these fronts along the axes are determined. If the thicknesses of the fluid layers are very small, then the second triangular front is converted into a back concave front, and a slow wave arises. This slow wave is of interest for seismology. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 190–211.     

On methods of deriving equations describing effective models of layered media

Methods of deriving equations describing effective models of layered periodic media are presented. Elastic and fluid media, as well as porous Biot media, may be among these media. First, effective models are derived by a rigorous method, and then some operations in the derivation are replaced by simpler ones providing correct results. As a consequence, a comparatively simple and justified method of deriving equations of an effective model is established. In particular, this method allows us to simplify to a degree and justify the derivation of an effective model for media containing Biot layers; this method also produces equations of an effective model of a porous layered medium intersected by fractures with slipping contacts. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998 pp. 219–243. Translated by L. A. Molotkov.     

Investigation of the wave field in an effective model of a layered elastic-fluid medium

For a medium that consists of alternating elastic and fluid layers, an effective model is constructed and investigated. This model is a special case of the Biot medium. The wave field is represented as Fourier and Mellin integrals. In the Mellin integral, the contour of integration is replaced by a stationary contour. In the expressions obtained, the order of integration is changed and the inner integral is calculated. The outer integral is equal to two residues. The corresponding poles are roots of two equations of fourth order. These roots lie on the right half-plane and may be complex or real. The representation obtained for the wave field is in agreement with the expressions derived by the Smirnov-Sobolev method. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 175–192.     

Sources of the center of dilatation and center of compression types in the Biot model

For a homogeneous isotropic model of porous Biot media, wave fields of spherically symmetric point sources are determined. The conditions under which a point source of the center of compression type can be replaced by two sources, one of which is a pair of oppositely directed forces and the other is a center of radially directed tangential forces, are obtained. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 196–213. Translated by L. A. Molotkov.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

Equations of an effective model of an anisotropic elastic medium with cracks filled with a fluid

For a periodic layered medium in which every period consists of an elastic anisotropic layer and a fluid homogeneous layer, an effective model is derived by averaging. This model describes wave propagation and has two phases. The equations of this model are deduced in the case of the general anisotropy and in some special cases. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 175–191. Translated by L. A. Molotkov.     

On attenuation in layered biot media and in effective models of them

Effective models of periodic layered porous Biot media possessing viscosity and relaxation are established and investigated. These models correspond to generalized Biot media with equations containing, as a rule, exponential kernels of relaxation and viscosity. Such kernels occur even if relaxation is absent in the initial medium. Inequalities that must be satisfield by the parameters of the kernels are established with the help of energy studies. Special cases in which the effective models possess no viscosity or no relaxation are stated. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 244–262. Translated by L. A. Molotkov     

On an effective model of an elastic block medium with slide contacts on the interfaces

The system under consideration consists of equal rectangular blocks. Each block contains four homogeneous elastic isotropic media with slide contacts on the interfaces between the media. For this system an effective four-phase model is deduced. By analyzing the equations of the effective model, the fronts and velocities along axes are determined. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 96–117. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by L. A. Molotkov.     

Algebraic bethe anzatz and the tavis-cummings model

The N-atom-radiation-field model is solved for its eigenvalues and eigenstates by the algebraic Bethe ansatz. The eigenenergies and the amplitudes of the wave functions are expressed in terms of the solutions of the Bethe equations. The determinant representations of the expectation values and the norms of the wave functions are obtained. The algebraic approach establishes a relationship between the model under consideration and the other exactly solvable models. Bibliography:12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 66–79. Translated by N. M. Bogolyubov.     

Existence,uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model

This paper is concerned with the existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model. We first establish the existence of traveling wave fronts by using geometric singular perturbation theory. Then the asymptotic behavior and uniqueness of traveling wave fronts are obtained by using the standard asymptotic theory and sliding method. In addition, our method is also suitable to establish the uniqueness and asymptotic behavior of traveling wave fronts for a cooperative system.     

Global stability and wavefronts in a cooperation model with state-dependent time delay

This article deals with a diffusive cooperative model with state-dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds. For the cooperative DDE system, the positivity and boundedness of solutions are firstly given. Using the comparison principle of the state-dependent delay equations obtained, the stability criterion of model is analyzed both from local and global points of view. When the diffusion is properly introduced, the existence of traveling waves is obtained by constructing a pair of upper–lower solutions and Schauder's fixed point theorem. Calculating the minimum wave speed shows that the wave is slowed down by the state-dependent delay. Finally, the traveling wavefront solutions for large wave speed are also discussed, and the fronts appear to be all monotone, regardless of the state dependent time delay. This is an interesting property, since many findings are frequently reported that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.     

On an effective model describing a layered periodic elastic medium with slide contacts on the interfaces

Effective models are derived for layered periodic elastic media'with slide contacts on all interfaces. In the case where each period consists of n layers with different plate velocities, the effective model has n phases. These models are investigated for typical media. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 192–212. Translated by L. A. Molotkov.     

Jumps of displacements and stresses as seismic sources in Biot's medium

Wave fields excited in a homogeneous isotropic Biot medium by point sources described in terms of discontinuities of displacements and stresses are determined. The results are represented in the form of relations involving Fourier-Bessel or Mellin integrals and in the form of Stokes-type formulas. The interrelations between these representations are established. Among all possible point sources exciting Biot's medium, the elementary sources, in terms of which any complicated linear source can be described, are selected. The special case where the wave fields in the two phases of the Biot medium are independent of one another is considered, and the corresponding sources in the Biot medium are compared with the known sources in elastic and fluid media.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 164–196.This work was supported by the Russian Foundation for Basic Research under grant No. 96-05-65904.     

On the ray method for Biot media

This paper is devoted to an investigation of wave propagation in a Biot porous medium, which consists of elastic and fluid phases. The space-time ray expansion of solutions of dynamic equations for a Biot medium is constructed (in the anisotropic inhomogeneous case). In the inhomogeneous isotropic case, a Rytov law analog is derived similarly to elasticity theory. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 112–131.     

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J ⁺ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.     

Traveling wave fronts in a delayed population model of Daphnia magna

This paper deals with the traveling wave fronts of a delayed population model with nonlocal dispersal. By constructing proper upper and lower solutions, the existence of the traveling wave fronts is proved. In particular, we show such a traveling wave front is strictly monotone.     

Wave propagation in an isolated porous Biot layer with closed pores on the boundaries

On the boundaries of such an isolated porous Biot layer, the total stresses and normal relative displacement are equal to zero. For this layer, the symmetric and antisymmetric dispersion equations are established and investigated. The wave field consists of normal waves. In this layer, one bending wave, two plate waves, and infinitely many normal waves propagate. For all these waves, we determine dispersion curves by analytical methods. The velocities of the bending wave and the second plate wave for the infinite frequency are equal to the Rayleigh velocity. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 173–189.     

Co-invasion waves in a reaction diffusion model for competing pioneer and climax species

In this paper, we consider a reaction diffusion model for competing pioneer and climax species. A previous work has established the existence of traveling wave fronts connecting two competition-exclusion equilibria in certain range of the parameters, while in this paper, we explore the possibility of traveling wave fronts connecting the pioneer-invasion-only equilibrium and the co-invasion equilibrium. By combining the Schauder’s fixed point theorem with a pair of the so called desired functions, we show that the model does support such co-invasion waves in some other ranges of parameters. We also determine the minimal speed for such co-invasion waves in terms of the parameters, and discuss some biological implications and significance of the results.     

On an effective two-phase model of a medium with cracks of finite length

An effective two-phase model of media with cracks filled with liquid is generalized to the case of finite cracks. Wave propagation in a half-space and in a free layer, described by the new model, is investigated. The results are compared with experimental data and with corresponding results in the case of media with infinite cracks. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 137–155. Translated by L. A. Molotkov.