Asymptotic properties of the solution of a delay system

We consider a differential-difference system with constant delay of special form. We show that the derivative of the solution of the system tends to zero by the exponential law. This fact is used to construct a numerical solution for large times. We give an example of numerical simulation for a two-dimensional system.     

A new family of first-order boundary conditions for the Maxwell system: derivation,well-posedness and long-time behavior

This work deals with the time-dependent Maxwell system in the case of TE-polarized electromagnetic waves, when associated with a family of first-order local boundary conditions. The boundary conditions are derived by using a micro-diagonalization method, actuated by the standard one of M.E. Taylor and involving pseudodifferential technics. The conditions differ from an arbitrary function and any of them leads to a well-posed mixed problem that is described by a continuous semi-group. The arbitrary function can be seen as a parameter and an asymptotic analysis in time shows that it can be chosen so that the resulting boundary condition is absorbing: the system is related to an energy functional that converges towards zero as time tends to infinity. By involving an invariant space for the Maxwell system, the limit state can be explicitly written as a solution to a boundary-value problem depending on the initial data. The long time behavior of the solution is then completely analyzed.     

Convergence of a Force‐Based Hybrid Method in Three Dimensions

We study a force‐based hybrid method that couples an atomistic model with the Cauchy‐Born elasticity model. We show that the proposed scheme converges to the solution of the atomistic model with second‐order accuracy, since the ratio between lattice parameter and the characteristic length scale of the deformation tends to 0. Convergence is established for the three‐dimensional system without defects, with general finite‐range atomistic potential and simple lattice structure. The proof is based on consistency and stability analysis. General tools for stability analysis are developed in the framework opseudodifference operators in arbitrary dimensions. © 2012 Wiley Periodicals, Inc.     

An asymptotical study of combinatorial optimization problems by means of statistical mechanics

The analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study. Simulated annealing, a metaheuristic for combinatorial optimization problems, is based on this analogy. In this paper we show how a statistical mechanics formalism can be utilized to analyze the asymptotic behavior of combinatorial optimization problems with sum objective function and provide an alternative proof for the following result: Under a certain combinatorial condition and some natural probabilistic assumptions on the coefficients of the problem, the ratio between the optimal solution and an arbitrary feasible solution tends to one almost surely, as the size of the problem tends to infinity, so that the problem of optimization becomes trivial in some sense. Whereas this result can also be proven by purely probabilistic techniques, the above approach allows one to understand why the assumed combinatorial condition is essential for such a type of asymptotic behavior.     

Asymptotic solutions of the one-dimensional linearized Korteweg–de Vries equation with localized initial data

The Cauchy problem with localized initial data for the linearized Korteweg–de Vries equation is considered. In the case of constant coefficients, exact solutions for the initial function in the form of the Gaussian exponential are constructed. For a fairly arbitrary localized initial function, an asymptotic (with respect to the small localization parameter) solution is constructed as the combination of the Airy function and its derivative. In the limit as the parameter tends to zero, this solution becomes the exactGreen function for the Cauchy problem. Such an asymptotics is also applicable to the case of a discontinuous initial function. For an equation with variable coefficients, the asymptotic solution in a neighborhood of focal points is expressed using special functions. The leading front of the wave and its asymptotics are constructed.     

Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities

We consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution can be arbitrary large. In this paper we derive an asymptotic formula of the solution, which characterizes the gradient blow-up of the solution in terms of conductivities of inclusions as well as the distance between inclusions. The asymptotic formula is expressed in bipolar coordinates in terms of the Lerch transcendent function, and it is valid for inclusions with arbitrary constant conductivities. We illustrate our results with numerical calculations.     

Nonsmooth Mapping of Linear Control Systems

This paper deals with linear control systems of a special form. The main goal of the paper is to find the exact analytic solution of the time-optimal control problem for an arbitrary linear control system with constant coefficients using the analytic solution of this problem for the canonical system. For this aim, we construct a certain nonsmooth mapping between the 0-controllability sets of the given systems. In other words, by means of this mapping, we investigate the equivalence of the systems with the same qualitative behavior in a neighborhood of the stationary point.     

一类时滞差分系统解的渐近性

本文对一类时滞差分系统给出了保证其有界解收敛的一些充分条件，所得结果推广和改进了一些文献的相关结果。     

Analytic solutions of unstable interfaces for all density ratios in axisymmetric flows

An interface between two fluids subject to an external force is hydrodynamically unstable. We extend the Layzer-type potential flow model for unstable interfaces to the system of finite density ratio in axially symmetric geometry and derive analytic solutions for growth rates of unstable interfaces over all times. The analytic expressions for bubble growth rates at finite times are given for arbitrary Atwood number. Predictions of the analytic solutions for growth rates are in excellent agreements with numerical results.     

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.     

Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions

This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution     

Finite-dimensional controller design for semilinear parabolic systems

This paper considers the stabilization of steady-state solutions of a semilinear parabolic system using finite-dimensional feedback controllers with support in an arbitrary open subset and which are active in one equation only. It is shown that such a controller, with dimension given by the largest algebraic multiplicity of the unstable eigenvalues of the linearized system, exponentially stabilizes the steady-state solution. An optimal design methodology for these types of controllers, which is based on the finite element approximation of the semilinear parabolic system, is introduced and illustrated by numerical simulation examples.     

一类具状态依赖时滞的微分系统解的渐近行为

研究了一类具有状态时滞的微分方程系统解的渐近行为,获得了该系统每一个有界解当t→∞时都趋于常向量,所获得的结果改进和扩展了已有文献的相关结果.     

The unsteady motion of a viscous fluid between rotating parallel walls

The motions of a viscous incompressible fluid, rotating initially as a rigid body with constant angular velocity together with parallel walls which bound it, acted upon by suddenly starting longitudinal oscillations of one of the walls, are investigated. The walls make an arbitrary angle with the axis of rotation. In general, the solution is obtained in the form of the sum of an infinite series and is represented by an integral containing an elliptic function. A number of special cass of the motion of the wall is considered. The results obtained are used to investigate certain structures of the boundary layers on the walls.     

Asymptotic expansion of the solution to the system of elasticity equations with a concentrated pulse force

For the linear system of elasticity equations, we consider the problem of wave excitation by a concentrated pulse force of arbitrary orientation. On assuming that the medium is isotropic and its density and elastic moduli are infinitely differentiable functions constant in some neighborhood of the source point, we write down an asymptotic series for the solution. The coefficients of the series determine the singular component of the solution, as well as the jumps, of the solution and its derivatives, as the characteristic cones are crossed, corresponding to longitudinal and transverse waves.     

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.     

Green's formula for systems of difference equations with constant coefficients

Systems of difference equations with constant coefficients are considered. A formula has been obtained which expresses an arbitrary solution in an arbitrary bounded grid region by the values of this solution at boundary points of the region.Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 615–622, May, 1969.     

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper,firstly,by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation,we construct parameterized delta-shock and constant density solutions,then we show that,as the flux perturbation vanishes,they converge to the delta-shock and vacuum state solutions of the zero-pressure flow,respectively.Secondly,we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure.Furthermore,we rigorously prove that,as the two-parameter flux perturbation vanishes,any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow;any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Finally,numerical results are given to present the formation processes of delta shock waves and vacuum states.     

Properties of the false vacuum as a quantum unstable state

We analyze properties of unstable vacuum states from the standpoint of quantum theory. Some suggestions can be found in the literature that some false (unstable) vacuum states can survive up to times when their survival probability takes a nonexponential form. At asymptotically large times, the survival probability as a function of the time t has an inverse power-law form. We show that in this time region, the energy of false vacuum states tends to the energy of the true vacuum state as 1/t ² as t→∞. This means that the energy density in the unstable vacuum state and hence also the cosmological constant Λ = Λ(t) should have analogous properties. The conclusion is that Λ in a universe with an unstable vacuum should have the form of a sum of the “bare” cosmological constant and a term of the type 1/t ²: Λ(t) ≡ Λ_bare + d/t ² (where Λ_bare is the cosmological constant for a universe with the true vacuum).