Spike-type contrast structures in reaction-diffusion systems

Stationary internal layers of spike type for multidimensional reaction-diffusion problems are considered. We prove the existence of internal layers of this type, construct their asymptotics, and investigate the stability of these stationary solutions. The consideration is based on an extension of the singular limit eigenvalue problem method. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 121–134, 2005.     

One‐dimensional approximations of the eigenvalue problem of curved rods

In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ? of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ? tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

For a system of n identical particles in a homogeneous magnetic field, the discrete spectrum of the Hamiltonian H^{α, m} on the subspaces of functions with permutational symmetry α and rotational (SO(2)) symmetry m is studied as m→∞. It is proved that the discrete spectrum of the operator H^α,m contains only one eigenvalue if certain conditions are satisfied. The asymptotic behavior of this eigenvalue as m→∞ is found. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 28–41, 1992. Translated by A. V. Lyakhovskaya     

Asymptotic behavior of the second eigenfrequency for a system of two bodies connected by a thin rod

The method of matched asymptotic expansions is applied to the construction of the second eigenvalue, which tends to zero, in the problem corresponding to a system of two bodies connected by a thin rod.Institute of Mathematics, Urals Branch of the Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 68–77, October, 1993.     

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     

The eigenvalue problem for solitary waves of a boussinesq equation,via a generalization of the rouché theorem

We consider the study of an eigenvalue problem obtained by linearizing about solitary wave solutions of a Boussinesq equation. Instead of using the technique of Evans functions as done by Pego and Weinstein in [R. Pego and M. Weinstein, Convective Linear Stability of Solitary Waves for Boussinesq equation. AMS, 99, 311–375] for this particular problem, we perform Fourier analysis to characterize solutions of the eigenvalue problem in terms of a multiplier operator and use the strong relationship between the eigenvalue problem for the linearized Boussinesq equation and the eigenvalue problem associated with the linearization about solitary wave solutions of a special form of the KdV equation. By using a generalization of the Rouché Theorem and the asymptotic behavior of the Fourier symbol corresponding to the eigenvalues problem for the Boussinesq equation and the Fourier symbol corresponding to the eigenvalues problem for the KdV equation, we show nonexistence of eigenvalues with respect to weighted space in a planar region containing the right-half plane.     

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.     

Extrapolation and superconvergence of the Steklov eigenvalue problem

On the basis of a transform lemma, an asymptotic expansion of the bilinear finite element is derived over graded meshes for the Steklov eigenvalue problem, such that the Richardson extrapolation can be applied to increase the accuracy of the approximation, from which the approximation of O(h ^3.5) is obtained. In addition, by means of the Rayleigh quotient acceleration technique and an interpolation postprocessing method, the superconvergence of the bilinear finite element is presented over graded meshes for the Steklov eigenvalue problem, and the approximation of O(h ³) is gained. Finally, numerical experiments are provided to demonstrate the theoretical results.     

On the asymptotic error estimate for a discrete problem of fourth-order accuracy

The leading term of the error of eigenvalues of a discrete analog of the eigenvalue problem for an elliptic operator with variable coefficients is obtained. A method for refining eigenvalues by evaluating a correction with the help of a discrete problem of second-order accuracy is proposed. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 153–160.     

Exact solution of a model of three bound oscillators with Stark nonlinearity

A model of three bound boson oscillators with Stark nonlinearity is introduced and solved by the quantum inverse scattering method. For the trilinear oscillator, the eigenvalue problem is reduced to the spectral problem for the second-order homogeneous differential equation. Bibliography: 11 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 122–128. Translated by B. M. Bekker.     

Concentration phenomena for neutronic multigroup diffusion in random environments

We study the asymptotic behavior of the principal eigenvalue of a weakly coupled, cooperative linear elliptic system in a stationary ergodic heterogeneous medium. The system arises as the so-called multigroup diffusion model for neutron flux in nuclear reactor cores, the principal eigenvalue determining the criticality of the reactor in a stationary state. Such systems have been well studied in recent years in the periodic setting, and the purpose of this work is to obtain results in random media. Our approach connects the linear eigenvalue problem to a system of quasilinear viscous Hamilton–Jacobi equations. By homogenizing the latter, we characterize the asymptotic behavior of the eigenvalue of the linear problem and exhibit some concentration behavior of the eigenfunctions.     

Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium

An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. The perturbation technique for solving the nonlinear eigenvalue problem is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law. It is shown that the eigenvalues of the nonlinear eigenvalue problem are fully determined by the exponents of the damage evolution law. In the paper the third-order (four-term) asymptotic expansions of the angular functions determining the stress and continuity fields in the neighborhood of the crack tip are given. The asymptotic expansions of the angular functions permit to find the closed-form solution for the problem considered.     

The Existence and Stability of Asymmetric Spike Patterns for the Schnakenberg Model

Asymmetric spike patterns are constructed for the two-component Schnakenburg reaction–diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. For a pattern with k spikes, the construction yields   k ₁  spikes that have a common small amplitude and   k ₂= k − k ₁  spikes that have a common large amplitude. A k -spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two interesting conclusions are that asymmetric patterns can exist for a reaction–diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O (1) time scale.     

Asymptotic Analysis of Spectral Properties of Finite Capacity Processor Shared Queues

We consider sojourn (or response) times in processor‐shared queues that have a finite customer capacity. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite capacity models where the system can only hold a large number K of customers. Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Airy equation. The dominant eigenvalue leads to the tail of a customer's sojourn time distribution. Some numerical results are given to assess the accuracy of the asymptotic results.     

Error bound on the eigenvalue of the difference analog of the spectral problem in a cylindrical coordinate system

We consider the eigenvalue problem for the operator defined in a rectangle whose vertical left-hand side coincides with the z-axis. A difference scheme is constructed by an integrointerpolation method. An error bound is obtained for the simple eigenvalue of the difference analog in weighted generalized spaces W ₂ ² and W ₂ ³ .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 29–35, 1986.     

Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials

Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.     

Asymptotic decay of solutions of the liouville equation under perturbations

We consider the Cauchy problem for a perturbed Liouville equation. An asymptotic solution is constructed with respect to the perturbation parameter by the two-scale expansion method; this construction can be applied over long time intervals. The main result is the definition of a deformation of the leading term of the asymptotic expansion within a slow time scale. Translated frommatematicheskie Zametki, Vol. 68, No. 2, pp. 195–209, August, 2000.     

Earth surface effects on active faults: An eigenvalue asymptotic analysis

We study in this paper an eigenvalue problem (of Steklov type), modeling slow slip events (such as silent earthquakes, or earthquake nucleation phases) occurring on geological faults. We focus here on a half space formulation with traction free boundary condition: this simulates the earth surface where displacements take place and can be picked up by GPS measurements. We construct an appropriate functional framework attached to a formulation suitable for the half space setting. We perform an asymptotic analysis of the solution with respect to the depth of the fault. Starting from an integral representation for the displacement field, we prove that the differences between the eigenvalues and eigenfunctions attached to the half space problem and those attached to the free space problem, is of the order of d^-2, where d is a depth parameter: intuitively, this was expected as this is also the order of decay of the derivative of the Green's function for our problem. We actually prove faster decay in case of symmetric faults. For all faults, we rigorously obtain a very useful asymptotic formula for the surface displacement, whose dominant part involves a so called seismic moment. We also provide results pertaining to the analysis of the multiplicity of the first eigenvalue in the line segment fault case. Finally we explain how we derived our numerical method for solving for dislocations on faults in the half plane. It involves integral equations combining regular and Hadamard's hypersingular integration kernels.     

Asymptotic expansion of the coefficients of radiating plane waves in the scattering problem on a smooth periodic boundary

High-frequency asymptotic expansion of the coefficients of radiating plane waves in the problem of grazing scattering of a plane wave on a smooth periodic boundary is derived. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 71–86, 1990. Translated by V. V. Zalipaev.