A mixed problem for the Laplace operator in a domain with moderately close holes

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ε and we define a perforated domain Ω_ε obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ε → 0. For ε small, we denote by u_ε the solution of a mixed problem for the Laplace equation in Ω_ε. We describe what happens to u_ε as ε → 0 in terms of real analytic maps and we compute an asymptotic expansion.     

The Validity of Generalized Ginzburg-Landau Equations

We consider parabolic systems defined on cylindrical domains close to the threshold of instability, in which the Fourier modes with positive growth rates are concentrated at a non-zero critical wave number. In particular, we consider systems for which a so-called Ginzburg–Landau equation can be derived. Due to the presence of continuous spectrum, classical bifurcation theory is not available to describe bifurcating solutions. Thus, we consider a modified system with artificial spectral gap, which possesses an infinite-dimensional centre manifold. The amplitude equation on this manifold is called a generalized Ginzburg–Landau equation. From previous work [18] it is known that the Fourier modes are exponentially concentrated at integer multiples of the critical wave number. Hence, the error made by this modification is exponentially small in powers of the bifurcation parameter. The approximations obtained via the generalized Ginzburg–Landau equation are valid on a much longer time scale than those obtained by using the classical Ginzburg–Landau equation as an amplitude equation.     

Convergence of a family of solutions to a Fujita-type equation in domains with cavities

The Dirichlet problem for a Fujita-type equation, i.e., a second-order quasilinear uniformly elliptic equation is considered in domains Ω_ε with spherical or cylindrical cavities of characteristic size ε. The form of the function in the condition on the cavities’ boundaries depends on ε. For ε tending to zero and the number of cavities increasing simultaneously, sufficient conditions are established for the convergence of the family of solutions {u _ε(x)} of this problem to the solution u(х) of a similar problem in the domain Ω with no cavities with the same boundary conditions imposed on the common part of the boundaries ?Ω and ?Ω_ε. Convergence rate estimates are given.     

Investigation of the spectrum of a model operator in a Fock space

We consider a model operator H corresponding to a quantum system with a nonconserved finite number of particles on a lattice. Based on an analysis of the spectrum of the channel operators, we describe the position of the essential spectrum of H. We obtain a Faddeev-type equation for the eigenvectors of H.     

One-parameter Family of Positive Solutions for a Nonlinear Integral Equation Arising in Physical Kinetics

The paper is devoted to the question of solvability of a Urysohn type nonlinear integral equation. This equation has an application in the kinetic theory of gases and can be derived from Boltzmann model equation. We prove an existence theorem of one-parameter family of positive solutions in the space of functions possessing linear growth at infinity. Moreover, for each member of this family we find an exact asymptotic formula at infinity. We obtain two-sided estimates for solution, as well as describe an iterative method for construction of solution.We conclude the paper by giving examples of functions that describe nonlinearity and satisfy the conditions of the main theorem.     

A conservative pressure-correction scheme for transient simulations of reacting flows

An algorithm is presented for numerical simulations of time-dependent low Mach number variable density flows with an arbitrary amount of scalar transport equations and a complex equation of state. The pressure-correction type algorithm is based on a segregated solution formalism. It is conservative and guarantees stable results, regardless of the difference in density between neighboring cells. Furthermore, states are predicted which exactly match the equation of state. In the one-dimensional example, considering non-premixed flames, a simplified flamesheet model is used to describe the combustion of fuel and oxidizer. We demonstrate that the predicted states exactly correspond to the equation of state. We illustrate the accuracy improvement due to higher order formulation and demonstrate grid convergence.     

A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k , where the order k is even unnecessary to be equal to the order n . In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study.     

The recirculating flow due to a moving lid on a cavity containing a Darcy–Brinkman medium

The lid-driven rectangular cavity containing a porous Darcy–Brinkman medium is studied. The governing equation is solved by an eigenfunction method which is much simpler than using biorthogonal series. It is found that the porous medium effect decreases both the strength and the number of recirculating eddies, especially for deep cavities.     

On the solution of boundary-value problems of Kolmogorov-Petrovskii-Piskunov type

We describe a number of properties of solutions of boundary-value problems for nonlinear ordinary differential equations of the same type as those studied by Kolmogorov, Petrovskii, and Piskunov in their well-known paper on waves described by the parabolic equation. We construct and justify the asymptotics of such solutions for large values of the modulus of the independent variable.     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

Nonlinear Nonstationary Schrödinger Equation for the Problem of a Bose–Einstein Condensate of Dilute Gases Interacting with an Electromagnetic Field

We derive the Schrödinger equation describing the coupling of the Bose–Einstein condensate of an ideal gas to an electromagnetic field. Its solutions allow finding the evolution of the radiation intensity and of populations of coherent atomic states with different values of the recoil momenta and can be used to describe a number of effects including light scattering and amplification, amplification of an atom beam (atom laser), and induced transparency.     

Well-posedness of a diffusive gyro-kinetic model

We study a finite Larmor radius model used to describe the ions distribution function in the core of a tokamak plasma, that consists in a gyro-kinetic transport equation coupled with an electro-neutrality equation. Since the last equation does not provide enough regularity on the electric potential, we introduce a simple linear collision operator adapted to the finite Larmor radius approximation. We next study the two-dimensional dynamics in the direction perpendicular to the magnetic field. Thanks to the smoothing effects of the collision and the gyro-average operators, we prove the global existence of solutions, as well as short time uniqueness and stability.     

The interaction of a magnetoelastic shear wave with longitudinal cavities in a conducting layer

We study the influence of a strong magnetic field on the interaction of a shear wave with longitudinal cylindrical cavities in an elastic ideally conducting layer. The resulting singular integral equation of the boundary-value problem under consideration is implemented numerically for the case of a single cavity. We present the results of computation of the stresses on the edge of a circular cavity and an elliptical cavity. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 74–78.     

Non-Equilibrium States of a Photon Cavity Pumped by an Atomic Beam

We consider a beam of two-level randomly excited atoms that pass one-by-one through a one-mode cavity. We show that in the case of an ideal cavity, i.e. no leaking of photons from the cavity, the pumping by the beam leads to an unlimited increase in the photon number in the cavity. We derive an expression for the mean photon number for all times. Taking into account leaking of the cavity, we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the total energy variation in both the ideal and the open cavities as well as the entropy production in the ideal cavity.     

Adiabatic Evolution Generated by a Schrödinger Operator with Discrete and Continuous Spectra

In the paper, we consider the one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. It is assumed that the corresponding stationary operator depending on time as a parameter has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. A solution close at some moment to an eigenfunction of the stationary operator is studied. We describe its asymptotic behavior in the case where the eigenvalues of the stationary operator move to the edge of the continuous spectrum and, having reached it, disappear one after another.     

Behavior of quasi-particles on hybrid spaces. Relations to the geometry of geodesics and to the problems of analytic number theory

We review our recent results concerning the propagation of “quasi-particles” in hybrid spaces — topological spaces obtained from graphs via replacing their vertices by Riemannian manifolds. Although the problem is purely classical, it is initiated by the quantum one, namely, by the Cauchy problem for the time-dependent Schrödinger equation with localized initial data.We describe connections between the behavior of quasi-particles with the properties of the corresponding geodesic flows. We also describe connections of our problem with various problems in analytic number theory.     

Global existence for weak solutions of the Cauchy problem in a model of radiation hydrodynamics

We consider a model based on the Navier–Stokes–Fourier system coupled to a transport equation, recently proposed in order to describe the thermal effects in low Mach number radiative flows. We establish global-in-time existence in weighted spaces for the associated Cauchy problem in the framework of weak solutions.     

Almost integrable evolution equations

We present a 2-component equation with exactly two nontrivial generalized symmetries, a counterexample to Fokas' conjecture that equations with as many symmetries as components are integrable. Furthermore we prove the existence of infinitely many evolution equations with finitely many symmetries. We introduce the concept of almost integrability to describe the situation where one has a finite number of symmetries. The symbolic calculus of Gel'fand-Dikiî andp-adic analysis are used to prove our results.     

Application of the transfer matrix approximation for wave propagation in a metafluid representing an acoustic black hole duct termination

The transfer matrix method has been proposed to analyze the acoustic black hole effect in duct terminations. The latter is achieved by placing a retarding waveguide structure inside the duct, which consists in a set of rings whose inner radii decrease to zero following a power law. The rings are separated by thin air cavities. If the number of ring-cavity ensembles is large enough, wave propagation inside the waveguide can be treated as a continuous problem. A governing differential equation can be derived for the acoustic black hole which intrinsically relies on assumptions from transfer matrix theory. However, no formal demonstration exists showing that the transfer matrix solution is consistent and formally tends to the solution of the continuous problem. Proving such consistency is the main goal of the paper and an original option has been adopted to this purpose. First, we prove the differential equation for the acoustic black hole is identical to the wave equation for a metafluid with a power-law varying density. Transfer matrices are then applied to solve wave propagation in a discretization of the metafluid into layers of constant density. It is shown that when the number of layers tends to infinity and their thicknesses to zero, the transfer matrix solution satisfies the metafluid equation and therefore the acoustic black hole one. The transfer matrices for the metafluid discrete layers take a particularly simple form, which is a great advantage. This work allows one to interpret the retarding waveguide structure as a particular realization of the metafluid.     

Time-dependent Schrödinger equation: Statistics of the distribution of Gaussian packets on a metric graph

We consider a time-dependent Schrödinger equation in which the spatial variable runs over a metric graph. The boundary conditions at the vertices of the graph imply the continuity of the function and the zero sum of the one-sided derivatives taken with some weights. In the semiclassical approximation, we describe a propagation of Gaussian packets on the graph that are localized at a point at the initial instant of time. The main focus is placed on the statistics of the behavior of asymptotic solutions as time increases. We show that the calculation of the number of quantum packets on a graph is related to the well-known number-theoretic problem of finding the number of integer points in an expanding simplex. We prove that the number of Gaussian packets on a finite compact graph grows polynomially. Several examples are considered. In a particular case, Gaussian packets are shown to be distributed on a graph uniformly with respect to the edge travel times.