Ray dynamics in a neighborhood of flat points of the boundary

The behavior of rays is studied in a neighborhood of boundary points where the curvature has a zero of multiplicity one (an inflection point) and multiplicity two (a flat point of a boundary which is concave from the side of the wave field). The rays considered away from the flat point of the boundary are connected with a whispering gallery wave incident on this point, and they are constructed on the basis of the known asymptotics of this wave. The results are represented in figures obtained with the help of the computer.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 146–155, 1981.     

Wave field in the caustic shadow in a neighborhood of an inflection point of the boundary

The problem of the propagation of whispering gallery waves in a neighborhood of an inflection point of the boundary is considered. It is shown that a caustic shadow zone occurs away from the boundary along the normal. The asymptotics of the wave field in the caustic shadow are obtained, and their geometric interpretation in terms of complex rays is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 246–260, 1979.     

Concentration of the solutions near a limit ray in the neighborhood of an inflection point of the boundary

One considers the question regarding the energy concentration of a wave field in the neighborhood of a limit ray in the problem of the behavior of whispering gallery waves near an inflection point of the boundary. An estimate related to this question is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 155–158, 1988.     

ON TRANSMISSION PROBLEM FOR KIRCHHOFF TYPE WAVE EQUATION WITH A LOCALIZED NONLINEAR DISSIPATION IN BOUNDED DOMAIN＊

In this article,we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials,one...     

On transmission problem for viscoelastic wave equation with a localized a nonlinear dissipation

In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials, one component being a Kirchhoff type wave equation with time dependent localized dissipation which is effective only on a neighborhood of certain part of boundary, while the other being a Kirchhoff type viscoelastic wave equation with nonlinear memory.     

Whispering-gallery waves in a neighborhood of an inflection point of the boundary. Asymptotics of the wave field as t→∞

The asymptotics is constructed as t for the wave field generated by a whispering-gallery wave incident on an inflection point of the boundary. The asymptotics has ray character and is connected with the family of rays tangent to the convex part of the boundary beyond the point of inflection.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 151–166, 1984.     

Stabilization of the Kirchhoff type wave equation with locally distributed damping

We derive energy decay estimates of the Kirchhoff type wave equation with a localized damping term in a bounded domain. The damping coefficient function may act alive only on a neighborhood of some part of the boundary.     

Short-wave scattering near the boundry inflection point

We investigate the problem of tangential incidence of short waves onto a surface with an inflection point. Formal solutions of the corresponding equation are constructed near the inflection point in the form of a quasihomogeneous function series. The formal solution is joined with the geometrical optics solution far from the inflection point of the boundary. The problem is restated as a scattering problem for the Schrodinger equation; existence, uniqueness, and smoothness theorems are proved. The formal asymptotic expansions are proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 152–166, 1985.In conclusion, I would like to thank M. M. Popov for suggesting the problem, and also V. M. Babich and M. M. Popov for useful comments.     

A note about positive solutions for an equation of Kirchhoff type

A nonlinear boundary value problem related to an equation of Kirchhoff is considered. The existence of positive solutions is proved through alternative Leray-Schauder’s type combined with Krasnoselskii’s fixed point theorem. Numerical methods are presented and a result of local convergence is established.     

Dual reciprocity hybrid radial boundary node method for the analysis of Kirchhoff plates

A meshless method of dual reciprocity hybrid radial boundary node method (DHRBNM) for the analysis of arbitrary Kirchhoff plates is presented, which combines the advantageous properties of meshless method, radial point interpolation method (RPIM) and BEM. The solution in present method comprises two parts, i.e., the complementary solution and the particular solution. The complementary solution is solved by hybrid radial boundary node method (HRBNM), in which a three-field interpolation scheme is employed, and the boundary variables are approximated by RPIM, which is applied instead of moving least square (MLS) and obtains the Kronecker’s delta property where the traditional HBNM does not satisfy. The internal variables are interpolated by two groups of symmetric fundamental solutions. Based on those, a hybrid displacement variational principle for Kirchhoff plates is developed, and a meshless method of HRBNM for solving biharmonic problems is obtained, by which the complementary solution can be solved.     

Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.     

Computation of the Second Variation of the Energy Functional of a Two-Phase Medium

A formula for the second variation of the energy functional ofa two-phase elastic medium is derived at a critical point of the functional. An estimate for the remainder is given. The critical field of displacement and the critical boundary of the interface of the phases are assumed to be sufficiently smooth. Computations are made inside a domain occupied by the elastic medium as well as in a neighborhood of the intersection of the boundary of this domai and the boundary of the interface of the phases. Bibliography: 6 titles.     

An optimization problem for the Biharmonic equation with Sobolev conditions

We state and solve an optimization problem about distribution of several supporting points under a Kirchhoff plate clamped along the boundary: the biharmonic equation is supplied with the Dirichlet boundary conditions and point Sobolev conditions. Some open questions are formulated. Bibliography: 23 titles.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

On flow switching bifurcations in discontinuous dynamical systems

In paper, the sliding dynamics on the separation boundary is discussed based on the set-valued vector field theory. From vector fields in the neighborhood of a specific separation boundary, the passability of the flow from the one domain into another one is further discussed. The switching bifurcation conditions from the passable boundary to the non-passable boundary are developed. The sliding flow fragmentation on the separation boundary surface is also presented. The normal vector product field function is introduced to determine the switching bifurcation and sliding fragmentation.     

Korteweg–de vries zero dispersion limit: Through first breaking for cubic-like analytic initial data

The characteristic solution of the initial value problem for the Riemann invariant form of the Korteweg-de Vries modulation equations is obtained locally through first breaking in the case of initial data which is analytic near a cubic inflection point. Tsarev's (see [11]) system is used to define the solution implicitly in terms of the speeds of the generalized KdV modulation equations. These “higher flow” speeds satisfy the same derivative identities obtained by Levermore for the KdV flow (see [6]). Moreover, by explicitly constructing the Lax-Levermore (see [5]) minimizer the Tsarev solution is shown to be the unique solution of Whitham's equations corresponding to the KdV zero dispersion limit. Also the Tsarev system possesses at most one three-sheeted solution in a uniform neighborhood of the inflection point. Finally the explicit minimizer provides phase information for the Venakides' small dispersion KdV waveform through first breaking; see [12]. © 1993 John Wiley & Sons, Inc.     

Boundary value problems of electroelasticity with concentrated singularities

We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighborhood of infinity or in the neighborhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.     

Behavior of the generalized solutions of the Dirichlet problem for higher-order elliptic equations in the neighborhood of the boundary

Making use of the method of weight functions and of energy inequalities, similar to the Saint-Venant principle, the authors obtain estimates which characterize the behavior of the generalized solutions of the Dirichlet problem for the general higher-order elliptic equation in the neighborhood of a boundary point. In the case of two independent variables one has obtained an estimate of the maximum of the modulus of the solution in the neighborhood of a boundary point.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 114–125, 1982.     

Existence and multiplicity of solutions for Kirchhoff type equations

In this paper, the existence and multiplicity of weak solutions are obtained for a class of Kirchhoff type problems with Dirichlet boundary value conditions by using the mountain pass theorem, the local linking theorem, the fountain theorem and the symmetric mountain pass lemma in critical point theory.     

Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.