On the Holmgren-John uniqueness theorem for the wave equation with piecewise analytic coefficients

In this paper a uniqueness theorem is proved for the wave equation in the domain Q^2T=Ω×(0,2T), where Ω is a piecewise analytic Riemannian manifold (Riemannian polyhedron). Initial data are assumed to be given on a part Γ₀ × (0, 2T) of the space-time boundary of the cylinder Q^2T, Γ₀. The uniqueness of a weak solution is proved “in the large,” in a domain formed by the corresponding characteristics of the wave equation. Bibliography:24 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 113–136. Translated by T. N. Surkova.     

The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains

We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 40–57, July, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01411.     

Elastic wave diffraction by a wedge-shaped inclusion

By means of Sommerfeld integrals techniques the problem of diffraction is reduced to a functional equation in a complex domain. Such an equation is known in the literature as an equation with non-Carleman translation. The theory of pseudodifferential operators reduces to the theory of functional equations and, consequently, the problem of diffraction, to a Fredholm integral equation. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 50–70, 1990. Translated by B. V. Budaev.     

Perturbation of a shock wave

The Cauchy problem is considered for the perturbed Hopf equation u_t+uu_x=εf(u), ε→0. The solution in the continuity domain can be expanded in the standard asymptotic series in integral powers of the small parameter. An asymptotic representation is found for the line of propagation of the shock wave. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 462–466, March, 1999.     

A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability

A boundary value problem for a second-order nonlinear singularly perturbed differential equation is considered for the case in which there is variation of stability caused by the intersection of roots of the degenerate equation. By the method of differential inequalities, we prove the existence of a solution such that the limit solution is nonsmooth. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 354–362, March, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00694.     

Excitation of a wave field in a triangular domain with impedance boundary conditions

The Helmholtz equation in a closed domain that is an equilateral triangle with inhomogeneous impedance boundary conditions is considered. A functional equation in which the unknown function is the Fourier-image of a wave field on the boundary of the domain is constructed. This functional equation is solved for the case of homogeneous boundary conditions (the problem on eigenvalues), as well as for the case of inhomogeneous boundary conditions in the absence of the resonance. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 300–318. Translated by A. V. Shanin.     

Scattering of a plane wave on a periodic curve with trapping domains

The problem of scattering on a periodic curve is considered. The asymptotic solution of the problem is constructed, and its principal terms are presented. The justification of the asymptotic solution found is provided. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 75–85. Translated by I. V. Kamotskii     

Diffraction of the plane wave by a transparent wedge: The Sommerfeld-Malyuzhinets approach

The time-harmonic diffraction of the plane wave by a transparent wedge of an arbitrary opening is examined by the Sommerfeld-Malyuzhinets approach. The velocities of wave propagation inside and outside the wedge are equal. An explicit solution is presented, which is studied asymptotically as kr → ∞ and kr → 0. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 157–172.     

Wood's anomalies in the scattering problem of a plane wave on a smooth periodic boundary

A series of multiple diffracted fields in a high-frequency asymptotic expansion of the solution to the problem of grazing scattering of a plane wave by a smooth periodic boundary is summed in the case of appearance of Wood's anomalies. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 87–100, 1990. Translated by V. V. Zalipaev.     

On the uniqueness of a solution of the inverse problem for a simple-layer potential

We prove the uniqueness of a solution of the inverse problem for a simple-layer potential for star-shaped smooth surfaces in the case of the metaharmonic equation Δv − K ² v = 0. For the Laplace equation, an analogous statement is not true. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 892–899, July, 2008.     

On the behavior near the caustic of a nonsteady wave field with singularity (A homogeneous generalized function) at the initial front

The Cauchy problem for the wave equation in the case of discontinuity on the initial front is investigated. The discontinuity is described by a homogeneous generalized function of degree λ. The transformation of the initial front while passing the space-time caustic is studied. The structure of the wave front and the space-time rays near the caustic is considered. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 122–133, 1990. Translated by N. Ya. Kirpichnikova.     

Justification of a construction of the solution of the problem on a point source ofSH waves in a layered medium with vertical boundary

The solution is determined by the method of separation of variables. The method is justified by using a suitable space of generalized functions. Boundary conditions “are hidden” in this space. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 56–66. Original article This work was supported by the Russian Foundation of Fundamental Research (Grant No. 93-011-16148). Translated by T. N. Surkova.     

The neumann problem for the wave equation in a cone

The Neumann problem for the wave equation in a wedge is considered. The asymptotic behavior of solutions to the problem in a neighborhood of the edge of the wedge is studied. In order to deduce and justify asymptotic formulas, the solvability of the problem in the scale of weight function spaces is investigated. Bibliography: 30 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 71–110.     

On the uniqueness of solutions of inverse multidimensional spectral problems in dual asymptotic models

The solution of an elliptic equation in an unbounded region is approximated by the solution of an ordinary differential equation. To obtain a new inverse problem we prove a uniqueness theorem and exhibit a solution algorithm. Bibliography: 14 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 132–140.     

On the case of the explicit integrability of the equation for SH-waves

The equation for SH-waves is considered for the following parameters μ and ρ: μ=a(x)·b(y), ρ=a(x)b(y)(c(x)+d(y)) (a,b,c,d are known functions). For such μ and ρ the variables in this equation can be separated. An explicit solution of the problem of the interaction of a whispering gallery wave with a vertical interface of two media is obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 12–16. Translated by N. S. Zabavnikova.     

Diffraction on a semi-infinite screen of a plane nonstationary wave the amplitude of which increases linearly along the front set

An exact analytic solution of the two-variables nonstationary problem of diffraction on an ideal half-infinite screen is obtained by the Smirnov-Sobolev method. The source of the field is an incident plane acoustic wave with a δ-function profile. The wave amplitude is a linear function increasing along the front set. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 138–152.     

Almost global wellposedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/ε] for initial data of the form ε Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,e ^T/ε ]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation. Financial support provided in part by NSF grant DMS-0400643.     

Yang-baxterization of the quantum dilogarithm

A new solution of the Yang-Baxter equation with spectral parameter is found. The resulting R-matrix R(x) is an operator inH⊗H, whereH=L ₂(ℝ). This R-matrix is required to justify the solution of the sine-Gordon model on the discrete space-time. Bibliography: 18 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 146–154. Translated by B. Bekker.     

Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence

We establish an estimate for the rate of convergence of a solution of an ordinary stochastic differential equation of order p ≥ 2 with a small parameter in the coefficient of the leading derivative to a solution of a stochastic equation of order p − 1 in the metric ρ(X, Y) = (sup_0≤t≤T M|X(t) − Y(t)|²)^1/2 __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1587–1601, December, 2006.     

Diffraction of a plane wave by a narrow-angle cone in the case of the dirichlet boundary conditions

By using Smyshlayev's formula for the amplitude of a wave scattered by the vertex of an arbitrary cone, an approximate formula for this amplitude is deduced for the case of a narrow-angle cone. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 3–11. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by N. S. Zabavnikova.