Exact local behavior of positive solutions for a semilinear elliptic equation with Hardy term

We characterize an exact growth order near zero for positive solutions of a semilinear elliptic equation with Hardy term. This result strengthens an existence result due to E. Jannelli [The role played by space dimension in elliptic critical problems, JDE 156 (1999), 407-426].

     

Ground states and spectrum of quantum electrodynamics of nonrelativistic particles

A system consisting of finitely many nonrelativistic particles bound on an external potential and minimally coupled to a massless quantized radiation field without the dipole approximation is considered. An ultraviolet cut-off is imposed on the quantized radiation field. The Hamiltonian of the system is defined as a self-adjoint operator in a Hilbert space. The existence of the ground states of the Hamiltonian is established. It is shown that there exist asymptotic annihilation and creation operators. Hence the location of the absolutely continuous spectrum of the Hamiltonian is specified.

     

The Hopf Bifurcation in a Parabolic Free Boundary Problem with Double Layers

We consider a parabolic free boundary problem which has a bifurcation parameter and double interfaces. We investigate the sign change in a real part of eigenvalues and the transversality condition as a bifurcation parameter cross the critical value in order to examine the stability of the stationary solutions. The occurence of a Hopf bifurcation will be shown at a critical value.     

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.     

Existence and uniqueness of generalized monopoles in six-dimensional non-Abelian gauge theory

In this paper, we establish the existence and uniqueness of the spherically symmetric monopole solutions in SO(5) gauge theory with Higgs scalar fields in the vector representation in six-dimensional Minkowski space-time and obtain sharp asymptotic estimates for the solutions. Our method is based on a dynamical shooting approach that depends on two shooting parameters which provides an effective framework for constructing the generalized monopoles in six-dimensional Minkowski space-time.     

A nontrivial example of application of the Nielsen fixed-point theory to differential systems: Problem of Jean Leray

In reply to a problem posed by Jean Leray in 1950, a nontrivial example of application of the Nielsen fixed-point theory to differential systems is given. So the existence of two entirely bounded solutions or three periodic (harmonic) solutions of a planar system of ODEs is proved by means of the Nielsen number. Subsequently, in view of T. Matsuoka's results in Invent. Math. (70 (1983), 319-340) and Japan J. Appl. Math. (1 (1984), no. 2, 417-434), infinitely many subharmonics can be generically deduced for a smooth system. Unlike in other papers on this topic, no parameters are involved and no simple alternative approach can be used for the same goal.

     

Stability of Taylor–Couette and Dean flow: A semi-analytical study

An alternative method is presented for solving the eigenvalue problem that governs the stability of Taylor–Couette and Dean flow. The eigenvalue problems defined by the two-point boundary value problems are converted into initial value problems by applying unit disturbance method developed by Harris and Reid [27] in 1964. Thereafter, the initial value problems are solved by differential transform method in series and the eigenvalues are computed by shooting technique. Critical wave number and Taylor number for Taylor–Couette flow are computed for a wide range of rotation ratio (μ), −4 ? μ ? 1 (first mode) and −2 ? μ ? 1 (second mode). The radial eigenfunction and cell patterns are presented for μ = −1, 0, 1. Also, we have computed critical wave number and Dean number successfully.     

Asymptotic Expansion for Layer Solutions of a Singularly Perturbed Reaction-Diffusion System

For a singularly perturbed -dimensional system of reaction--
diffusion equations, assuming that the 0th order solutions possess boundary and internal layers and are stable in each regular and singular region, we construct matched asymptotic expansions for formal solutions in all the regular, boundary, internal and initial layers to any desired order in . The formal solution shows that there is an invariant manifold of wave-front-like solutions that attracts other nearby solutions. We also give conditions for the wave-front-like solutions to converge slowly to stationary solutions on that manifold.

     

Elliptic equations with jumping nonlinearities involving high eigenvalues

We present some multiplicity results concerning semilinear elliptic Dirichlet problems with jumping nonlinearities where the jumping condition involves a set of high eigenvalues not including the first one. Using a variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. Indeed, we prove that for every positive integer $k$ there exists a positive integer $n(k)$ such that, if the number of jumped eigenvalues is greater than $n(k),$ then the problem has at least a solution which presents $k$ peaks. Moreover, we describe the asymptotic behaviour of the solutions as the number of jumped eigenvalues tends to infinity; in particular, we analyse some concentration phenomena of the peaks (near points or suitable manifolds), we describe the asymptotic profile of the rescaled peaks, etc $\ldots $      

On the Asymptotic and Numerical Analyses of Exponentially III-Conditioned Singularly Perturbed Boundary Value Problems

Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [1], is used to analytically calculate high-order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [2]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are shown to compare very favorably with two-term asymptotic results. Finally, some Sturm-Liouville operators with exponentially small spectral gap widths are studied. One such problem is applied to analyzing metastable internal layer motion for a certain forced Burgers equation.     

Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities

In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.).     

Angular self-intersections for closed geodesics on surfaces

In this note we consider asymptotic results for self-intersections of closed geodesics on surfaces for which the angle of the intersection occurs in a given arc. We do this by extending Bonahon's definition of intersection forms for surfaces.

     

The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method, which assumes perturbed Jacobians at each step. Its high convergence orders were characterized by Dennis and Moré [Math. Comp. 28 (1974), 549-560]. The inexact Newton method constitutes another such model, since it assumes that at each step the linear systems are only approximately solved; the high convergence orders of these iterations were characterized by Dembo, Eisenstat and Steihaug [SIAM J. Numer. Anal. 19 (1982), 400-408]. We have recently considered the inexact perturbed Newton method [J. Optim. Theory Appl. 108 (2001), 543-570] which assumes that at each step the linear systems are perturbed and then they are only approximately solved; we have characterized the high convergence orders of these iterates in terms of the perturbations and residuals.

In the present paper we show that these three models are in fact equivalent, in the sense that each one may be used to characterize the high convergence orders of the other two. We also study the relationship in the case of linear convergence and we deduce a new convergence result.

     

Interfacial instabilities and fingering formation in Hele-Shaw flow

The interfacial instability of Hele-Shaw flow has been a crucialissue for the understanding of the pattern formation of viscousfingers in a Hele-Shaw cell. By using a unified asymptotic approach,we derive two different types of instability mechanisms for‘slightly’ time-dependent finger solutions; namely,(i) the global-trapped-wave (GTW) instability; and (ii) thezero-frequency (null-f) instability. On the basis of these instabilitymechanisms, the selection of viscous finger formation is clarified;the apparent contradiction between the previous linearstabilityanalysis by Tanveer (1987, Phys. Fluid 30, 1589) and othersand the numerical simulations by DeGregoria & Schwartz (1986,J. Fluid Mech. 164, 383)and the experimental evidence is reconciled.     

On exponential ill-conditioning and internal layer behavior

In the limit of small difTusivity the internal layer behavior associated with the initial-boundary value problems for a viscous shock equation and a reaction diffusion equation is analyzed.As a result of the occurrence of exponentially small eigenvalues for the linearized problems the steady state internal layer solutions are shown to very sensitive to small perturbations.For the time dependent problems the small eigenvalues give rise to exponentially slow internal layer motion.Accurate numerical methods are used to compute the steady state internal layer solutions and the slow internal layer motion.The relationship between the viscous shock problem and some exponentially ill-conditioned linear singular perturbation problems is discussed.     

Discrete approximation of integral operators

A method to approximate the eigenvalues of linear operators depending on an unknown distribution is introduced and applied to weighted sums of squared normally distributed random variables. This area of application includes the approximation of the asymptotic null distribution of certain degenerated U- and V-statistics.

     

Nonlinear waves in complex oscillator network with delay

We investigate the spatio-temporal patterns of Hopf bifurcating periodic solutions in a delay complex oscillator network. Firstly, we calculate the critical values of Hopf bifurcation. Secondly, the bifurcating periodic solutions can take on two cases: one is synchronization or anti-synchronization, and another is the coexistence of two phase-locked, N mirror-reflecting and N standing waves, because the system has group symmetry. Finally, the stability of these nonlinear oscillations is determined using the center manifold theorem and normal form method with the imaginary eigenvalues being simple and double.     

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the k th and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.     

Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source

We consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source. We find the genus-zero algebraic function describing the limit mean density of eigenvalues in the case of an anharmonic potential and a diagonal external source with two symmetric eigenvalues. We discuss critical regimes where the density support changes the connectivity or increases the genus of the algebraic function and consequently obtain local universal asymptotic representations for the density at interior and boundary points of its support (in the generic cases). The investigation technique is based on an analysis of the asymptotic properties of multiple orthogonal polynomials, equilibrium problems for vector potentials with interaction matrices and external fields, and the matrix Riemann-Hilbert boundary value problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 34–57, April, 2009.     

On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping

This paper is concerned with the asymptotic behavior of solutions of the critical generalized Korteweg-de Vries equation in a bounded interval with a localized damping term. Combining multiplier techniques and compactness arguments it is shown that the problem of exponential decay of the energy is reduced to prove the unique continuation property of weak solutions. A locally uniform stabilization result is derived.