Exact Solutions for the Flow of Fractional Maxwell Fluid in Pipe-Like Domains

This paper presents an analysis of unsteady flow of incompressible fractional Maxwell fluid filled in the annular region between two infinite coaxial circular cylinders. The fluid motion is created by the inner cylinder that applies a longitudinal time-dependent shear stress and the outer cylinder that is moving at a constant velocity. The velocity field and shear stress are determined using the Laplace and finite Hankel transforms. Obtained solutions are presented in terms of the generalized G and R functions. We also obtain the solutions for ordinary Maxwell fluid and Newtonian fluid as special cases of generalized solutions. The influence of different parameters on the velocity field and shear stress is also presented using graphical illustration. Finally, a comparison is drawn between motions of fractional Maxwell fluid, ordinary Maxwell fluid and Newtonian fluid.     

Complete analytic solution of the geodesic equation in Schwarzschild-(anti-)de Sitter spacetimes

The complete set of analytic solutions of the geodesic equation in a Schwarzschild-(anti-)de Sitter space-time is presented. The solutions are derived from the Jacobi inversion problem restricted to the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The solutions are completely classified by the structure of the zeros of the characteristic polynomial which depends on the energy, angular momentum, and the cosmological constant.     

On the polarizability of a pair of cylinders in a transverse electric field

A sequential scheme for calculating the polarizability of a pair of parallel cylinders with an arbitrary (but fairly symmetric) form is suggested. An infinite set of algebraic equations is derived for coefficients involved in an expression for the potential. The numerical solution of this set makes it possible to find the polarizability with any degree of accuracy. This method is unrelated to the cylinder shape. The electrostatic properties of the cylinders are described in terms of the multipolar polarizability matrix.     

Response of infinite length rods and beams with periodically varying area

This paper develops a solution method for the longitudinal motion of a rod or the flexural motion of a beam of infinite length whose area varies periodically. The conventional rod or beam equation of motion is used with the area and moment of inertia expressed using analytical functions of the longitudinal (horizontal) spatial variable. The displacement field is written as a series expansion using a periodic form for the horizontal wavenumber. The area and moment of inertia expressions are each expanded into a Fourier series. These are inserted into the differential equations of motion and the resulting algebraic equations are orthogonalized to produce a matrix equation whose solution provides the unknown wave propagation coefficients, thus yielding the displacement of the system. An example problem of both a rod and beam are analyzed for three different geometrical shapes. The solutions to both problems are compared to results from finite element analysis for validation. Dispersion curves of the systems are shown graphically. Convergence of the series solutions is illustrated and discussed.     

A Jost-Schroer theorem for string fields

It is shown that the truncated Wightman functions of three or more string-localized fields vanish if they are solutions of a Klein-Gordon equation in each variable. As an application it is shown that a string field is a free field if its two-point functions are those of a free field. Another application to perturbation theory is pointed out.     

Solution of a mixed parity nonlinear oscillator: Harmonic balance

The method of harmonic balance is used to calculate first-order approximations to the periodic solutions of a mixed parity nonlinear oscillator. First, the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction. Then the two auxiliary equations, where the restoring force functions are odd, are solved by using a first harmonic term (without a constant). Therefore, we obtain the two approximate solutions to the mixed parity nonlinear oscillator. One is expressed in terms of the exact amplitude in the negative direction, the other in terms of the approximate amplitude. These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincaré method for large amplitudes.     

Multiple scattering from gyrotropic bianisotropic cylinders of arbitrary cross sections using the modeling technique

Based on the principle of the equivolumetric model, the multiple scattering from two-dimensional composite gyrotropic bianisotropic cylinders (GBC's) possessing a certain magnetic group of symmetry is studied by using the indirect modeling technique. At first, a mathematical formulation for the problem is carried out and its solution expressed in terms of a system of simultaneous linear equations of infinite order. The normal incident wave can be either TE(z) or TM(z) polarization. The linear equation system is then numerically solved by truncating the infinite series. Its numerical results are obtained and presented to characterize the co- and cross-polarized scattering by some typical composite GBC's. Especially, the multiple interactions among all the modeling cylinders, i.e., the host and guest cylinders as well as the guest and guest cylinders, are examined. Some phenomena of the interactions are found and discussed.     

第二类变系数KdV方程的新类型无穷序列精确解

为了构造变系数非线性发展方程的无穷序列新精确解, 发掘第一种椭圆辅助方程的构造性和机械化性特点, 获得了该方程的 新类型解和相应的 Bäcklund 变换. 在符号计算系统 Mathematica 的帮助下, 以第二类变系数 KdV 方程为应用实例, 构造了三种类型的无穷序列新精确解. 这里包括无穷序列光滑类孤子解、无穷序列尖峰孤立子解和无穷序列紧孤立子解. 这种方法也可以获得其他变系数非线性发展方程的无穷序列新精确解.     

Electrostatic and optical resonances of arrays of cylinders

The solutions of the electrostatic potential problem for the square and hexagonal arrays of circular cylinders with zero applied field (homogeneous or resonant solutions) are studied. We show that for non-touching cylinders, the set of resonances is discrete except in the neighbourhood of one point, at which the dielectric constant of the array has an essential singularity. For arrays of touching cylinders, the set is well represented by a continuous distribution. This representation enables the derivation of the asymptotic form of the expansion for the dielectric constant of the array when the dielectric constant of the cylinders is large. The known value of the first term in the expansion enables us to derive the second term. The physical characteristics of the resonant solutions are studied. Metals achieve values of dielectric constant which are close to the resonant values (real and negative) for certain wavelengths. Curves are given which enable the prediction of those wavelengths at which the optical resonances of both arrays occur, for any area fraction and composition of a columnar cerment film.     

Exact analytical results for a two-level quantum system under a Lorentzian-shaped pulse field

We investigate a two-level quantum system driven by a Lorentzian-shaped pulse field. An analytical solution is presented in terms of the confluent Heun functions. It is shown that for specially chosen parameter conditions, there are a number of the exact analytical solutions in an explicit form. The dependence of the final transition probabilities in the two levels on the system parameters is derived analytically and confirmed numerically.     

Two-state model of a general Heun class with periodic level-crossings

A specific constant-amplitude periodic level-crossing model of the semi-classical quantum time-dependent two-state problem that belongs to a general Heun class of field configurations is presented. The exact analytic solution for the probability amplitude, generally written for this class in terms of the general Heun functions, in this specific case admits series expansion in terms of the incomplete Beta functions. Terminating this series results in an infinite hierarchy of finite-sum closedform solutions each standing for a particular two-state model, which generally is only conditionally integrable in the sense that for these field configurations the amplitude and phase modulation functions are not varied independently. However, there exists at least one exception when the model is unconditionally integrable, that is the Rabi frequency and the detuning of the driving optical field are controlled independently. This is a constant-amplitude periodic level-crossing model, for which the detuning in a limit becomes a Dirac delta-comb configuration with variable frequency of the levelcrossings. The exact solution for this model is derived, the Floquet exponents are determined and study of the population dynamics in the system for various regions of the input parameters is done.     

Painlevé Analysis,Soliton Collision and Bäcklund Transformation for the (3+1)-Dimensional Variable-Coefficient Kadomtsev-Petviashvili Equation in Fluids or Plasmas

In this paper, we investigate a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlevé analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials, bilinear form and soliton solutions are obtained, and Bäcklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.     

Computation of the homogeneous and forced solutions of a finite length, line-driven, submerged plate

A formulation is developed to predict the vibration response of a finite length, submerged plate due to a line drive. The formulation starts by describing the fluid in terms of elliptic cylinder coordinates, which allows the fluid loading term to be expressed in terms of Mathieu functions. By moving the fluid loading term to the right-hand side of the equation, it is considered to be a force. The operator that remains on the left-hand side is the same as that of the in vacuo plate: a fourth-order, constant coefficient, ordinary differential equation. Therefore, the problem appears to be an inhomogeneous ordinary differential equation. The solution that results has the same form as that of the in vacuo plate: the sum of a forced solution, and four homogeneous solutions, each of which is multiplied by an arbitrary constant. These constants are then chosen to satisfy the structural boundary conditions on the two ends of the plate. Results for the finite plate are compared to the infinite plate in both the wave number and spatial domains. The theoretical predictions of the plate velocity response are also compared to results from finite element analysis and show reasonable agreement over a large frequency range.     

A new algorithm for anisotropic solutions

We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only.     

Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients

The analytical nonautonomous rogons are reported for the inhomogeneous nonlinear Schrödinger equation with variable coefficients in terms of rational-like functions by using the similarity transformation and direct ansatz. These obtained solutions can be used to describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, and Bose-Einstein condensates, respectively. Moreover, the snake propagation traces and the fascinating interactions of two nonautonomous rogons are generated for the chosen different parameters. The obtained nonautonomous rogons may excite the possibility of relative experiments and potential applications for the rogue wave phenomenon in the field of nonlinear science.     

Free vibration of rectangular plates of arbitrary thickness

Free vibration of thick rectangular plates is investigated by using the “method of initial functions” proposed by Vlasov. The governing equations are derived from the three-dimensional elastodynamic equations. They are obtained in the form of series and theories of any desired order can be constructed by deleting higher terms in the infinite order differential equations. The numerical results are compared with those of classical, Mindlin, and Lee and Reismann solutions.     

A study of correlation functions for the delta-function Bose gas

The Gel'fand-Levitan equation for the quantum nonlinear Schrödinger field theory is used to investigate the correlation functions of the delta-function Bose gas. Operator expressions are derived for the field and for nonlocal products of fields in terms of the quantized reflection operators which create and annihilate eigenstates of the Hamiltonian. For the two-point function, an explicit series expression is obtained in which the nth term is determined by well-defined n-body combinatorics in an infinite volume. The inductive properties of this series are discussed and used to express the temperature and chemical potential dependence of the correlation functions entirely in terms of previously known thermodynamic functions. The zero separation limit of the series for the two-point function reproduces the thermodynamics derived by Yang and Yang, while the infinite coupling limit gives the Fredholm determinant result of Schultz and Lenard. The latter is related to the Painlevé V equation by the monodromy arguments of Jimbo, Miwa, Mori, and Sato. The (1/c) correction to the large coupling limit is calculated from the Gel'fand-Levitan series and expressed in terms of solutions to Painlevé V. The asymptotic behavior of the relevant Painlevé function is discussed and related to the long range behavior of the correlation function.     

Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect

In this paper, a set of 3D general solutions to static problems of 1D hexagonal piezoelectric quasicrystals is obtained by introducing two displacement functions and utilizing the rigorous operator theory. All the physical quantities are expressed by five quasi-harmonic functions. Based on the general solutions and with the help of the superposition principle, fundamental solutions for infinite/half-infinite spaces are presented by trial-and-error technique. The general solutions can be conveniently used to solve the boundary value problems regarding dislocations, cracks and inhomogeneities. The fundamental solutions are of primary significance to development of numerical codes such as boundary element method.     

Closed-form solution for a pair of touching cylindrical conductors in an external electric field

We give expressions for the electric potential and field around two conducting cylinders in contact, in closed form and in terms of elementary functions. The polarizability tensor, and hence the torque acting on the cylinder pair, follow from these results. The surface charge density, and the total charge per unit length on each cylinder, can also be evaluated in closed form. The equal and opposite forces on the two cylinders are found in the case of equal cylinder radii.     

Light scattering by dielectric particles with axial symmetry: II

A new method is developed for calculating optical characteristics of axially symmetric particles. Electromagnetic fields are separated into two (axisymmetric and non-axisymmetric) parts. The light scattering problem is formed in the integral form and solved independently for each of the parts by using specially chosen scalar potentials. The potentials are expanded into series in spherical wave functions, and the expansion coefficients are calculated from solving the infinite systems of linear algebraic equations. The applicability of the proposed method for solving the problem of light scattering by Chebyshev particles, spheroids, and finite circular cylinders is briefly discussed, and some results of calculations performed for these particles are presented.