Localized Asymptotic Solution of the Wave Equation with a Radially Symmetric Velocity on a Simplest Decorated Graph

In the present paper, we obtain a complete asymptotic series for a solution of the Cauchy problem for a wave equation with variable velocity on the simplest decorated graph obtained by gluing a ray to the Euclidean space \(\mathbb{R}^3\). It is assumed here that the localized initial conditions are given on the ray, and the velocity on \(\mathbb{R}^3\) is radially symmetric. The energy distributions are also described as the small parameter tends to zero.     

Localized asymptotic solutions of the wave equation with variable velocity on the simplest graphs

The asymptotic behavior of the Cauchy problem for the wave equation with variable velocity and localized initial conditions on the line, semi-axis, and an infinite starlike graph is described. The solution consists of a short-wave and long-wave parts; the shortwave part moves along the characteristics, while the long-wave part satisfies the Goursat or Darboux problem. In the case of a star-like graph, the distribution of energy with respect to the edges is discussed; this distribution depends on the arrangement of the eigensubspaces of the unitary matrix that defines the boundary condition at the vertex of the star.     

The Canopy Graph and Level Statistics for Random Operators on Trees

For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended ‘canopy graph.’ For this tree graph, the random Schrödinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular – pure point possibly with singular continuous component which is proven to occur in some cases.     

Asymptotic solutions of the Cauchy problem with localized initial conditions for linearized two-dimensional Boussinesq-type equations with variable coefficients

In the paper, asymptotic solutions of the Cauchy problem with localized initial data for the two-dimensional wave equation (with variable speed) which is also perturbed by (spatially) variable weakly dispersive components are constructed. We consider both the case of normal dispersion occurring in the linearized Boussinesq equation for water waves over smoothly changing bottom and the case of anomalous dispersion arising when studying the wave equation with rapidly oscillating velocity. With regard to the fact that the front of the solution has focal points and self-intersection points, we present formulas based on the modified Maslov canonical operator in the case of initial perturbations of a rather general form which decrease at infinity. For perturbations of special form, we express the asymptotic behavior of a solution in the vicinity of the front, using derivatives of the sum of squares of the Airy functions Ai and Bi.     

Solution of a cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula

The Cauchy problem for a class of diffusion equations in a Hilbert space is studied. It is proved that the Cauchy problem in well posed in the class of uniform limits of infinitely smooth bounded cylindrical functions on the Hilbert space, and the solution is presented in the form of the so-called Feynman formula, i.e., a limit of multiple integrals against a gaussian measure as the multiplicity tends to infinity. It is also proved that the solution of the Cauchy problem depends continuously on the diffusion coefficient. A process reducing an approximate solution of an infinite-dimensional diffusion equation to finding a multiple integral of a real function of finitely many real variables is indicated.     

E-Polarized Scalar Wave Diffraction by Perfectly Conductive Arbitrary Shaped Cylindrical Obstacles with Finite Thickness

The Analytical Regularization Method is applied to the problem of E-polarized wave diffraction by arbitrary shaped cylindrical obstacle which is perfectly conductive and homogeneous in longitudional direction. The initial electromagnetic boundary value problem is reduced to the infinite algebraic system of the second kind by means of constructed analytical regularization procedure. As it is well-known, an equation of such a kind, in principal, can be solved with any predetermined accuracy by means of truncation procedure. Numerical implementation of corresponding analytical regularization procedure is suggested. Numerical results thus obtained show high quality of the algorithm, including relatively small values and uniform boundness of condition number of truncated algebraic systems when their size tends to infinity. By the qualitative property of infinite algebraic system of the second kind, it guarantees the stability of numerical process of truncated solving as well as the convergence of solution of truncated system to the solution of infinite system. Relevant numerical results, including condition number behaviour, current density, field space distribution and far field pattern for a few different resonant and non-resonant obstacles are presented.     

Singular Hartree equation in fractional perturbed Sobolev spaces

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.     

On a class of solutions of nonlinear Boltzmann equations

In a recent paper by Krook and Wu, the nonlinear Boltzmann equation for an infinite, spatially homogeneous, isotropic monoatomic gas of constant density and kinetic energy and with an elastic differential cross section that varies inversely as relative speed has been reduced to an infinite sequence of moment equations. The present note observes that the moment equations are successively integrable and shows that as time goes to infinity, the distribution tends to be Maxwellian.     

Fractional Number Operator and Associated Fractional Diffusion Equations

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.     

A theoretical and numerical resolution of an acoustic multiple scattering problem in three-dimensional case

This paper develops an analytical solution for sound, electromagnetic or any other wave propagation described by the Helmholtz equation in three-dimensional case. First, a theoretical investigation based on multipole expansion method and spherical wave functions was established, through which we show that the resolution of the problem is reduced to solving an infinite, complex and large linear system. Second, we explain how to suitably truncate the last infinite dimensional system to get an accurate stable and fast numerical solution of the problem. Then, we evaluate numerically the theoretical solution of scattering problem by multiple ideal rigid spheres. Finally, we made a numerical study to present the “Head related transfer function” with respect to different physical and geometrical parameters of the problem.     

The Vlasov–Poisson System with Infinite Mass and Energy

This paper deals with solutions to the Vlasov–Poisson system with an infinite mass. The solution to the Poisson equation cannot be defined directly because the macroscopic density is constant at infinity. To solve this problem, we decompose the solution to the kinetic equation into a homogeneous function and a perturbation. We are then able to prove an existence result in short time for weak solutions to the equation for the perturbation, even though there are no a priori estimates by lack of positivity.     

Behavior of separatrix of one static model at infinity

A method for investigating the equations of static models describing the scattering of a relativistic particle on a fixed center in projective spaces is proposed. The basis of the method is a construction of a set of affine coordinates of the problem at zero total energy. This method is exemplified by the three-row crossing-symmetry matrix. Each element of the set results in a solution with the same Riemann surface on which the behavior of the separatrix at infinity is considered.     

Optimal Decay Estimates on the Linearized Boltzmann Equation with Time Dependent Force and their Applications

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L ^p -L ^q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n ≥ 3, the global existence and decay rates of solutions to the Cauchy problem are obtained under the condition that the force and source decay in time with some rates. This time decay restriction can be removed for space dimension n ≥ 5. Moreover, the existence and asymptotic stability of the time periodic solution are given for space dimension n ≥ 5 when the force and source are time periodic with the same period.     

Eigentensors of the Lichnerowicz operator in Euclidean Schwarzschild metrics

Properties of the eigentensors of the Lichnerowicz Laplacian for the Euclidean Schwarzschild metric are discussed together with possible applications to the linear stability of higher‐dimensional instantons. The main statement of the article is that any eigentensor of the Lichnerowicz operator in a Euclidean (possibly higher‐dimensional) Schwarzschild metric is essentially singular at infinity.     

Trilocal structures. V. Wave equations

In addition to the usual centroid-time wave equation, a trilocal structure will need to satisfy two relative-time wave equations. When the trilocal wave function is expanded in tree functions, each of the three wave equations becomes an infinite matrix equation, but when the four auxiliary conditions (defined in earlier articles in this series) are introduced, each wave equation reduces to a set of 16 linear homogeneous equations in 16 unknown expansion coefficients (the first 16 coefficients in the tree expansion). The 48 linear equations, in the 16 unknownC _j , are given explicitly. Every 16-by-16 determinant, formed from any 16 of these 48 linear homogeneous equations, must vanish if the trilocal structure is to be an acceptable solution; this requirement will be used in later calculations.     

Axially symmetric vibrations of fluid-filled poroelastic circular cylindrical shells

Employing Biot's theory of wave propagation in liquid saturated porous media, axially symmetric vibrations of fluid-filled and empty poroelastic circular cylindrical shells of infinite extent are investigated for different wall-thicknesses. Let the poroelastic cylindrical shells are homogeneous and isotropic. The frequency equation of axially symmetric vibrations each for a pervious and an impervious surface is derived. Particular cases when the fluid is absent are considered both for pervious and impervious surfaces. The frequency equation of axially symmetric vibrations propagating in a fluid-filled and an empty poroelastic bore, each for a pervious and an impervious surface is derived as a limiting case when ratio of thickness to inner radius tends to infinity as the outer radius tends to infinity. Cut-off frequencies when the wavenumber is zero are obtained for fluid-filled and empty poroelastic cylindrical shells both for pervious and impervious surfaces. When the wavenumber is zero, the frequency equation of axially symmetric shear vibrations is independent of nature of surface, i.e., pervious or impervious and also it is independent of presence of fluid in the poroelastic cylindrical shell. Non-dimensional phase velocity for propagating modes is computed as a function of ratio of thickness to wavelength in absence of dissipation. These results are presented graphically for two types of poroelastic materials and then discussed. In general, the phase velocity of an empty poroelastic cylindrical shell is higher than that of a fluid-filled poroelastic cylindrical shell.The phase velocity of a fluid-filled bore is higher than that of an empty poroelastic bore. Previous results are shown as a special case of present investigation. Results of purely elastic solid are obtained.     

Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of the MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of the MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.     

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

We construct the formal solution to the Cauchy problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrödinger operator for a quantum particle in a time-dependent potential.     

Finite-amplitude electrostatic waves im magneto-active plasmas

Summary Weakly nonlinear dispersive longitudinal waves in an infinite homogeneous collisionless plasma in the presence of an external constant and uniform magnetic field are considered. Under specific physical assumptions and for an arbitrary three-dimensional envelope modulation of a plane wave, a purely differential system is derived. Taking into account the effect of wave-wave and wave-particle interaction, the evolution of the modulation is described by a modified nonlinear Schr?dinger equation, coupled to the space perturbation charge densities. The generation of a static mode is described. As a particular case the electron waves are discussed and some special solutions, resorting to the theory of the perturbed solitions.     

Regularity of Solutions to Vorticity Navier–Stokes System on ℝ<Superscript>2</Superscript>

The Cauchy problem for the Navier–Stokes system for vorticity on plane is considered. If the Fourier transform of the initial data decays as a power at infinity, then at any positive time the Fourier transform of the solution decays exponentially, i.e. the solution is analytic.