Vibrations of double-string complex system under moving forces. Closed solutions

In this paper the dynamic response of a double-string system traversed by a constant or a harmonically oscillating moving force is considered. The force is moving with a constant velocity on the top string. The strings are identical, parallel, one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a double-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case a part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations.     

LS解法和Fisher方程行波系统的定性分析

提出了求解非线性发展方程的新方法——LS解法.LS解法是基于(G’/G)展开法和扩展的双曲正切函数展开法.并引入了Poincar定性理论的思想,然后以Fisher方程为例进行了试验.通过定性分析首先获得了Fisher方程行波系统积分曲线的性质,然后解得了Fisher方程作为耗散系统时单调减少的波前解和作为扩张系统时单调递增的波前解.一些试验结果与Ablowitz所得结果一致.也得到了Fisher方程作为扩张系统时的新结果.LS解法是在定性理论指导下,在已获知解曲线性质的情况下进行精确求解的,求解目标明确.LS解法揭示了线性系统也可以用作辅助方程来求解非线性系统.     

具有一般非线性弹性力和广义阻尼力的相对转动非线性系统的周期解问题

讨论了一类相对转动非线性动力系统的周期解问题.首先建立了一类具有一般非线性弹性力、广义阻尼力和强迫周期力项的相对转动非线性动力系统;其次得到了对应自治系统的周期解不存在性结果,以及运用Mawhin重合度理论得到了该模型的周期解存在性结果,推广了已有的结果;最后举例证明本文结果的正确性.     

Obtaining the bidirectional transmittance distribution function of isotropically scattering materials using an integrating sphere

This paper demonstrates a method to determine the bidirectional transmittance distribution function (BTDF) using an integrating sphere. Information about the sample’s angle-dependent scattering is obtained by making transmittance measurements with the sample at different distances from the integrating sphere. Knowledge about the illuminated area of the sample and the geometry of the sphere port in combination with the measured data combines to a system of equations that includes the angle-dependent transmittance.The resulting system of equations is an ill-posed problem which rarely gives a physical solution. A solvable system is obtained by using Tikhonov regularization on the ill-posed problem. The solution to this system can then be used to obtain the BTDF.Four bulk-scattering samples were characterised using two goniophotometers and the described method to verify the validity of the new method. The agreement shown is excellent for the more diffuse samples. The solution to the low-scattering samples contains unphysical oscillations, but still gives the correct shape of the solution. The origin of the oscillations and why they are more prominent in low-scattering samples are discussed.     

Extension of Operational Matrix Technique for the Solution of Nonlinear System of Caputo Fractional Differential Equations Subjected to Integral Type Boundary Constrains

We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of our approach is the most simple orthogonal polynomials. Several new matrices are derived that have strong applications in the development of computational scheme. The scheme presented in this article is able to convert nonlinear coupled system of FDEs to an equivalent S-lvester type algebraic equation. The solution of the algebraic structure is constructed by converting the system into a complex Schur form. After conversion, the solution of the resultant triangular system is obtained and transformed back to construct the solution of algebraic structure. The solution of the matrix equation is used to construct the solution of the related nonlinear system of FDEs. The convergence of the proposed method is investigated analytically and verified experimentally through a wide variety of test problems.     

Solution space of discrete Wiener-Hopf equations and Grassmann manifold

It is shown that the solution space of a system of discrete Wiener-Hopf equations is a set of points on an infinite-dimensional Grassmann manifold. Fractional transformations acting on the solution space are also discussed.     

Painleve Analysis and the Yang-Mills Equation in Two Specific Cases

Two important configurations of Yang-Mills fields are discussed applying to them the Painleve test of Ablowitz. The case with a constant electric E-field does not admit an exact solution while the cylindrically symmetric self-dual system admits an inverse scattering approach for instanton solution.     

Spatiotemporal measurement using L-band ESR-CT system for water sonolysis in the presence of 1-Hydroxy-2,2,5,5-tetramethyl-3-imidazoline-3-oxide

When using ESR to measure the radicals generated by ultrasound, it is necessary to extract a solution and place it in the ESR system. To avoid this process, we incorporated an ultrasonic reaction cell in an L-band ESR-CT system, producing a system that allows the detection of the concentration of radicals during ultrasonic irradiation. This system was used to measure the time and space dependences of OH radicals generated by ultrasonic irradiation. When a 10 ml aqueous solution of 1-hydroxy-2,2,5,5-tetramethyl-3-imidazoline-3-oxide (HTIO) was irradiated with ultrasound, it was found that the generation of radicals was clearly shown in a CT image after a period of 10 min. It was also found that continued irradiation resulted in an increased concentration of radicals. In addition to this system, an X-band ESR system was also used to measure the concentration of OH radicals generated, and the results of both systems were then compared. Both results are very similar, showing that the proposed system, which was realized by incorporating an ultrasonic irradiation cell in the L-band ESR-CT system, operated properly. Because this system allows the measurement of sonochemical reactions in an opaque cell or an opaque solution such as blood and industrial wastewater, it is a very useful measurement system for achieving the applying of sonochemistry to the medical engineering field.     

对称圆环状多原子分子的自由振动

考虑对称的圆环状多原子分子的振动问题,给出了该振动系统简正频率的数学表达式和自由振动的一般解.     

Calculation of short-range order in the method of cluster components

A scheme is proposed for calculating the short-range order in the method of cluster components for an n-component solid solution, taking into account an arbitrary number of spheres. The calculation of a single-, two-, and three-coordination spheres for a binary system and a calculation of a single sphere for a triple system are considered in detail. In the case of a binary system the possible type of property-composition dependence for a random distribution of the atoms in the case of one, two, and three coordination spheres are analyzed. The method is illustrated using a calculation of the concentration dependence of the elasticity constants of a KI_cBr_1-c solid solution.     

Solitons in collisionless cold plasma

The relativistic nonlinear self-consistent equations for a collisionless plasma with stationary ions are transformed into a form appropriate for finding exact analytic solutions. It is shown that for an axial system with planar geometry, the two-dimensional stationary equations for this system can be reduced to the sh-Gordon equation. The exact solution of this equation describing the charge-density equilibrium configuration is obtained, the solution having sharp transverse boundaries and a soliton form in longitudinal direction. The generalization to the nonstationary case is considered in an perturbative approach.     

Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle

We consider the Hamiltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subjected to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition, which is a version of the “Fermi Golden Rule.” We prove that in the large time approximation, any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.     

The pulse response of non-linear systems

The response of a non-linear, non-conservative, single degree of freedom system subjected to a pulse excitation is analysed. A transformation of the displacement variable is effected. The transformation function chosen is the solution of the linear problem subjected to the same pulse. With this transformation the equation of motion is brought into a form where Anderson's ultraspherical polynomial approximation is applicable for the solution of the problem. The method is applied to a cubic Duffing oscillator subjected to various pulses. The pulses considered are cosine, exponentially decaying and the step function. The analytical results are compared with the digital solution obtained on an IBM 360/344 system by using a Runge-Kutta fourth order method. The analytical results compare well with the digital solution.     

Collective excitations of harmonically trapped ideal gases

We theoretically study the collective excitations of an ideal gas confined in an isotropic harmonic trap. We give an exact solution to the Boltzmann-Vlasov equation; as expected for a single-component system, the associated mode frequencies are integer multiples of the trapping frequency. We show that the expressions found by the scaling ansatz method are a special case of our solution. Our findings are most useful in case the trap contains more than one phase: we demonstrate how to obtain the oscillation frequencies in case an interface is present between the ideal gas and a different phase.     

An Analytical Self-consistent Solution for the Free Energy of a 1-D Chain of Atoms Including Anharmonic Effects

The self-consistent method of lattice dynamics (SCLD) is used to obtain an analytical solution for the free energy of a periodic, one-dimensional, mono-atomic chain accounting for fourth-order anharmonic effects. For nearest-neighbor interactions, a closed-form analytical solution is obtained. In the case where more distant interactions are considered, a system of coupled nonlinear algebraic equations is obtained (as in the standard SCLD method) however with the number of equations dramatically reduced. The analytical SCLD solutions are compared with a numerical evaluation of the exact solution for simple cases and with molecular dynamics simulation results for a large system. The advantages of SCLD over methods based on the harmonic approximation are discussed as well as some limitations of the approach.     

Multi-hump bright solitons in a Schrödinger–mKdV system

We consider the problem of energy transport in a Davydov model along an anharmonic crystal medium obeying quartic longitudinal interactions corresponding to rigid interacting particles. The Zabusky and Kruskal unidirectional continuum limit of the original discrete equations reduces, in the long wave approximation, to a coupled system between the linear Schrödinger (LS) equation and the modified Korteweg–de Vries (mKdV) equation. Single- and two-hump bright soliton solutions for this LS–mKdV system are predicted to exist by variational means and numerically confirmed. The one-hump bright solitons are found to be the anharmonic supersonic analogue of the Davydov's solitons while the two-hump (in both components) bright solitons are found to be a novel type of soliton consisting of a two-soliton solution of mKdV trapped by the wave function associated to the LS equation. This two-hump soliton solution, as a two component solution, represents a new class of polaron solution to be contrasted with the two-soliton interaction phenomena from soliton theory, as revealed by a variational approach and direct numerical results for the two-soliton solution.     

Infinite series symmetry reduction solutions to the modified KdV--Burgers equation

From the point of view of approximate symmetry, the modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV--Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV--Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.     

A geometric wave function for a few interacting bosons in a harmonic trap

We establish a new geometric wave function that combined with a variational principle efficiently describes a system of bosons interacting in a one-dimensional trap. By means of a combination of the exact wave function solution for contact interactions and the asymptotic behavior of the harmonic potential solution we obtain the ground state energy, probability density and profiles of a few boson system in a harmonic trap. We are able to access all regimes, ranging from the strongly attractive to the strongly repulsive one with an original and simple formulation.     

3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis

We prove by an explicit construction that solutions to incompressible 3D Euler equations defined in the periodic cube Ω=[0,L]³ can be mapped bijectively to a new system of equations whose solutions are globally regular. We establish that the usual Beale-Kato-Majda criterion for finite-time singularity (or blowup) of a solution to the 3D Euler system is equivalent to a condition on the corresponding regular solution of the new system. In the hypothetical case of Euler finite-time singularity, we provide an explicit formula for the blowup time in terms of the regular solution of the new system. The new system is amenable to being integrated numerically using similar methods as in Euler equations. We propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite-time singularity problem of Euler equations, based on the conservation of quantities directly related to energy and circulation. The method of mapping to a regular system can be extended to any fluid equation that admits a Beale-Kato-Majda type of theorem, e.g. 3D Navier-Stokes, 2D and 3D magnetohydrodynamics, and 1D inviscid Burgers. We discuss briefly the case of 2D ideal magnetohydrodynamics. In order to illustrate the usefulness of the mapping, we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1D inviscid Burgers equation.     

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.