Axially symmetric solutions in general scalar-tensor theory

We generalise Ernst's derivation of the axially symmetric solutions of Einstein's field equations to the general scalar-tensor theory proposed by Nordtvedt. The solution of the Nordtvedt theory differs by a conformai transformation from the Brans-Dicke solution. The Kerr-like solution of the Nordtvedt theory is obtained as an example.     

Static conformally flat solutions in a general scalar-tensor theory of gravitation

Explicit field equations in the general scalar-tensor theory of gravitation proposed by Nordtvedt are obtained with the aid of a static spherically symmetric conformally flat metric. Exact static solutions of Nordtvedt-Barker field equations both in vacuum and in the presence of a source-free electromagnetic field are presented and studied. It is shown that there are no spherically symmetric static conformally flat solutions of Nordtvedt-Barker field equations representing perfect fluid distribution with disordered radiation obeying the equation of state=3p, except for the trivial empty flat space-time of Einstein's theory.     

The Post-Newtonian Effects of the Rotation of the Central Body on the Motion of the Celestial Body in Three Gravitational Theories

The post-Newtonian effects of the rotation of the central body on the variation of celestial orbital elements are studied according to the post-Newtonian metric theory. The variation of celestial orbital elements caused by the rotation of the central body in three gravitational theories of Einstein, Brans-Dicke and Nordtvedt is obtained by using the method of general perturbation. The resulting effects are the periodic variation of inclination, eccentricity and mean anomaly; the periodic and secular variation of longitudes of periastron and ascending node and mean longitudes of epoch, but the semimajor axis remains unperturbed (no variation). In addition, the obtained theoretical results are applied to the calculation of the post-Newtonian effect of the rotation of the sun on the variation of the orbital elements of planets in solar system. The numerical results are given in nble 1. Finally, the obtained results are discussed and compared with other theories.     

Two-Soliton Solutions of Axially Symmetric Metrics

Stationary axially symmetric solutions of Einstein's field equations generated by the soliton technique are presented in this paper with Laplace's solution as seed. The solutions are asymptotically flat and the Schwarzschild, Kerr and Kerr-NUT metrics are contained in it. The constructed solutions possess an event horizon. The surface area of the event horizon is evaluated. Finally, the solutions presented in this paper are compared with the solutions of Manko. A simple transformation technique is discussed by which one can directly obtain the solutions of Gutsunaev and Manko simply by adjusting a parameter related to the Inverse Scattering Method.     

Exactly solvable time-dependent models in quantum mechanics and their applications

Methods for obtaining exact and approximate solutions of the evolution of quantum-mechanical problems are discussed. The cyclic evolution of quantum systems described by time-periodic Hamiltonians is analyzed. A class of time-periodic Hamiltonians is constructed in the close analytical form. The corresponding cyclic solutions are calculated. Time-dependent Hamiltonians are generated whose expectation values calculated with cyclic solutions are time independent. It is shown that the expectation values of the spin projection calculated with the same cyclic solutions, as well as the probability density of finding a particle at a given space-time point, are also time independent. Therefore, the approach can be used to simulate quantum dynamic potential wells with the particle localization effect. Nonadiabatic geometric phases are expressed in terms of the cyclic solutions. Exactly solvable time-dependent problems are used to construct a universal set of gates for quantum computers. A method for obtaining entanglement operators is discussed.     

Transient motions of bi-lattices

We develop formal solutions for the propagation of transient pulses on a variety of bi-lattice models. The lattices are composed of a finite homogeneous chain connected in series with a different semi-infinite homogeneous chain at a common location occupied by a single mass which is different from the masses of both chains. Exact analytic solutions of this general case are not possible. Some analytic solutions are, however, possible for a variety of special cases. The general solutions are illustrated by numerically inverting the Laplace transform functions. The exact solutions are found to correlate very well with the numerical inversion scheme. Such correlations give confidence in the numerical scheme's predictions of the solutions of the more complicated chains.     

Lorentz-violating effects on topological defects generated by two real scalar fields

The influence of a Lorentz violation on soliton solutions generated by a system of two coupled scalar fields is investigated. Lorentz violation is induced by a fixed tensor coefficient that couples the two fields. The Bogomol’nyi method is applied and first-order differential equations are obtained whose solutions minimize the energy and are also solutions of the equations of motion. The analysis of the solutions in phase space shows how the stability is modified with the Lorentz violation. It is shown explicitly that the solutions preserve linear stability despite the presence of Lorentz violation. Considering Lorentz violation as a small perturbation, an analytical method is employed to yield analytical solutions.     

Poloidal–toroidal decomposition in a finite cylinder: II. Discretization,regularization and validation

The Navier–Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non-axisymmetric helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective.     

Single-scale two-dimensional-three-component generalized-Beltrami-flow solutions of incompressible Navier-Stokes equations

Generalized-Beltrami-flow (GBF) solutions, which are exact solutions of incompressible Navier-Stokes equations (NSE), are still rare. Most existing GBF solutions are either planar or axisymmetric cases. We derive analytically a series of single-scale two-dimensional-three-component (2D3C) GBF solutions under the framework of helical decomposition. These solutions yield a manifold of fixed points with infinite degrees of freedom in the solution space. The key of the derivation is to arbitrarily put different wave vectors at the same wave length, and to apply a novel parallel relation to any pair of these wave vectors. Although these solutions belong to a general class of 2D3C Euler solutions, to our knowledge there has been no publication focusing on these particular GBF forms. The significance of these GBF solutions is that the novel parallel relation implies new statistical relations on turbulence energy transfer and velocity phases.     

A new experimental approach to Mach's principle and relativistic graviation

As noted many years ago by Sciama, and more recently by Nordtvedt, Lorentz invariant (relativistic) gravitation at linear order involves a vector potential that is required to properly account for large inertial effects as well as the correct prediction of the classical tests of general relativity theory (GRT). It is pointed out that the linear-order vector aspect of the gravitational potential makes possible a simple, powerful and inexpensive technique for testing the predictions of GRT and associated issues. An experiment using this technique gives preliminary results that, to order of magnitude, corroborate GRT.1. If one demands a theory that satisfies Mach's Principle irrespective of the particular value of _c, one must go to a theory that contains GRT with critical cosmic matter density as a special case. Such a theory (an Einstein-Cartan theory with teleparallelism) has been developed by Treder [2].2. Equation (4) here is Nordtvedt's Eq. (14), in Ref. 3, with GRT PPN parameters chosen.3. The exact value of this correction factor that depends on the way in which energy is distributed between field and sources in turn depends on how the source term for the gravitational field equations is constructed. At least two different source terms that give correct predictions for the various tests of GRT exist. In this connection see Peters [4]. This ambiguity does not mean that it is impossible in principle to determine how energy is distributed between sources and field. Indeed, if one posits the existence of critical cosmic matter density, this experiment can decide the issue.     

Static,Spherically Symmetric Solutions of Yang-Mills-Dilaton Theory

Static, spherically symmetric solutions of the Yang-Mills-Dilaton theory are studied. It is shown that these solutions fall into three different classes. The generic solutions are singular. Besides there is a discrete set of globally regular solutions further distinguished by the number of nodes of their Yang-Mills potential. The third class consists of oscillating solutions playing the role of limits of regular solutions, when the number of nodes tends to infinity. We show that all three sets of solutions are non-empty. Furthermore we give asymptotic formulae for the parameters of regular solutions and confront them with numerical results.     

Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio

In this paper we analyse convective solutions of a two dimensional fluid layer in which viscosity depends exponentially on temperature. This problem takes in features of mantle convection, since large viscosity variations are to be expected in the Earth’s interior. These solutions are compared with solutions obtained at constant viscosity. Special attention is paid to the influence of the aspect ratio in the solutions presented. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, systematic method for gaining insight into the possible stationary solutions satisfied by the basic equations. One feature presented by the fluid with non constant viscosity is the presence of pitchfork and saddle-node subcritical bifurcations and the presence of convective solutions below the linear critical threshold. The analysis also provides limits of existence of stationary solutions and draws the boundaries for time dependent convection.     

Multi-cut solutions of Laplacian growth

A new class of solutions to Laplacian growth (LG) with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut LG solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

Poloidal–toroidal decomposition in a finite cylinder: II. Discretization, regularization and validation

The Navier–Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non-axisymmetric helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective.     

Asymptotic Behavior of Soliton Solutions with a Double Spectral Parameter for Principal Chiral Field

The soliton solutions with a double spectral parameter for the principal chiral field are derived by Darboux transformation. The asymptotic behavior of the solutions as time tends to infinity is obtained and the speeds of the peaks in the asymptotic solutions are not constants.     

Analytical and numerical solutions of the density dependent diffusion Nagumo equation

In this Letter, we obtained solutions to a class of density dependent diffusion Nagumo equations. In particular, series solutions are obtained, along with a bound for the range of the convergence. Also, numerical solutions are obtained by the Runge-Kutta-Fehlberg 45 method. Moreover, the dependence of the traveling wave solutions on various parameters is discussed. Furthermore, we compare the series solutions with the numerical solutions to validate the numerical method. The results obtained in this study reveal many interesting behaviors that warrant further study on the Nagumo equation.     

Asymptotic Behavior of Soliton Solutions with a Double Spectral Parameter for Principal Chiral Field

The soliton solutions with a double spectral parameter for the principal chiral field are derived by Darboux transformation. The asymptotic behavior of the solutions as time tends to infinity is obtained and the speeds of the peaks in the asymptotic solutions are not constants.     

Analytical Solutions for the Two-Dimensional Gross-Pitaevskii Equation with a Harmonic Trap

Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation,a nonlinear Schrdinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential.The approximate analytical solutions are obtained successfully.Comparisons between the analytical solutions and the numerical solutions have been made.The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.     

Asymptotic solutions of the inhomogeneous Burgers equation

The method of the active second harmonic suppression in resonators is investigated in this paper both analytically and numerically. The resonator is driven by a piston which vibrates with two frequencies. The first one agrees with an eigenfrequency and the second one is equal to the two times higher eigenfrequency. The phase shift of the second piston motion is 180 deg. It is known that for this case it is possible to describe generation of the higher harmonics by means of the inhomogeneous Burgers equation. This model equation was solved for stationary state analytically by a number of authors but only for ideal fluids. Unlike their solutions, new asymptotic solutions are presented here which take into account dissipative effects. The asymptotic solutions are compared with numerical ones. For study of generation higher harmonics the solutions are developed in a spectral form.