Simple Exact Solution Research Articles

Exact solutions to the equations of ideal gas-dynamics by means of the substitution principle

In this paper the equations governing three-dimensional unsteady flows of an inviscid, thermally non-conducting fluid, subjected to no extraneous force, are considered. First, some simple exact solutions are explicitly determined; then, by using the invariance of the given equations with respect to a generalized stretching group of all but space variables and a generalized time translation new solutions are immediately constructed. This invariance is known in the literature as substitution principle, since it allows to generate new solutions (substituted flows) from given solutions.

An exact solution of the kompaneets equations for a strong point explosion in a medium with quadratic density decrease

In the Kompaneets approximation, we study the propagation of a shock front from the explosion point in an inhomogeneous medium with a power-law variation of the density characterized by the exponent n=2, which is important and corresponds to the external parts of the solar and stellar coronas. An unexpectedly simple exact solution, which allows us to clarify the structure of the general solution under an arbitrary monotonous law of the density variation, was obtained. The results for the plane case are compared with Korikanskii’s results for a noncentral point explosion in a radially stratified medium. A correspondence between the exact solutions is established in these cases, and a new solution for the noncentral explosion is obtained for the case of a density with a singularity at a finite radius, which decreases quadratically at infinity.

Inflationary cosmology and thermodynamics

We present a simple and thermodynamically consistent cosmology with a phenomenological model of quantum creation of radiation due to vacuum decay. Thermodynamics and Einstein's equations lead to an equation in which H is determined by the particle number N. The model is completed by specifying the particle creation rate , which leads to a second-order evolution equation for H. We propose a simple that is naturally defined and that conforms to the thermodynamical conditions: (i) the entropy production rate starts at a maximum; (ii) the initial vacuum (for radiation) is a non-singular regular vacuum and (iii) the creation rate is initially higher than the expansion rate H, but then falls below H. The evolution equation for H then has a remarkably simple exact solution, in which a non-adiabatic inflationary era exits smoothly to the radiation era, without a reheating transition. For this solution, we give exact expressions for the cosmic scale factor, energy density of radiation and vacuum, temperature, entropy and super-horizon scalar perturbations.

The quasi-steady state cosmology: Some recent developments

This paper summarises the recent work on the quasi-steady state cosmology. This includes, the theoretical formulation and simple exact solutions of the basic equations, their relationship to observations, the stability of solutions and the toy model for understanding the growth of structures in the universe.

On a Family of Axially Symmetric Kinematically Determined Plastic Flows

For axially symmetric flows of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming either a plastic potential or dilatant double shearing, a set of three first-order partial differential equations is obtained. These equations turn out to be deceptive, because although they are simple in appearance, the determination of simple exact solutions is nontrivial. For one of the known families of solutions, the authors present a simple and straightforward asymptotic expansion for the stress angle and show that in a restricted range, this expansion provides an accurate solution of the nonlinear ordinary differential equation. In addition, for the same family of solutions, the authors examine a special case that permits one integration and enables the velocity components to be given explicitly in terms of the stress angle. This partial analytical solution may then be coupled either with the asymptotic expansion or with a full numerical solution of the governing first-order differential equation for the stress angle. In addition, the authors show that a special case of the nonlinear ordinary differential equation admits a simple first integral that has not been given previously. Finally, a detailed numerical comparison is made for various parameter values of the asymptotic expansion, the full numerical solution, and with known exact analytical solutions.

EXACT SOLUTIONS OF CYCLICALLY SYMMETRIC OSCILLATOR EQUATIONS WITH NON-LINEAR COUPLING. PART III: GLOBALLY STABLE SINGLE-FREQUENCY LIMIT CYCLES

Several authors have found very simple exact limit cycle solutions in certain sets of N cyclically symmetric coupled non-linear oscillator equations. In part I of this paper it was shown that the search for sets of equations with similar solutions is made much simpler if the equations are expressed in terms of the normal modes of the unperturbed (linear) system. In various applications it is desirable that the equations have a limit cycle solution with a single frequency and that this solution be globally stable. It is found here that the transformation of the equations to the unperturbed normal modes can greatly simplify the construction of such systems.

EXACT SOLUTIONS OF CYCLICALLY SYMMETRIC OSCILLATOR EQUATIONS WITH NON‐LINEAR COUPLING. PART III: GLOBALLY STABLE SINGLE‐FREQUENCY LIMIT CYCLES

Several authors have found very simple exact limit cycle solutions in certain sets of N cyclically symmetric coupled non-linear oscillator equations. In part I of this paper it was shown that the search for sets of equations with similar solutions is made much simpler if the equations are expressed in terms of the normal modes of the unperturbed (linear) system. In various applications it is desirable that the equations have a limit cycle solution with a single frequency and that this solution be globally stable. It is found here that the transformation of the equations to the unperturbed normal modes can greatly simplify the construction of such systems. © 1997 by John Wiley & Sons, Ltd.

The velocity equations for dilatant granular flow and a new exact solution

For axially symmetric flow of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming dilatant double shearing a set of three first order partial differential equations are obtained. These equations turn out to be deceptive because although simple in appearance, the determination of simple exact solutions is non-trivial. Here we show that all the known functional forms of existing solutions also arise systematically by consideration of the "optimal systems" of the classical Lie symmetries which indicates that any further solution types most likely arise from non-classical symmetries. For one of the known families we present a special case which admits a particularly simple closed form expression, which has not been previously given in the literature. For this particular special case the integral curves (streamlines) can be readily obtained as well as a simple analytical "approximation" for the particle paths. The streamlines and the validity of the analytical approximation are shown graphically.

N=2 extremal black holes.

It is shown that extremal magnetic black hole solutions of N = 2 supergravity coupled to vector multiplets $X^\Lambda$ with a generic holomorphic prepotential $F(X^\Lambda)$ can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon. A simple exact solution is found for the special case that the ratios of the $X^\Lambda$ are real, and it is seen that the logarithm of the conformal factor of the spatial metric equals the Kahler potential on the vector multiplet moduli space. Several examples are discussed in detail.

Exact solutions of cyclically symmetric oscillator equations with non‐linear coupling. Part II: Coupling with phase shift

AbstractSeveral authors have found very simple exact limit cycle solutions in cyclically symmetric systems of N oscillator equations with linear coupling in zero order of a perturbation parameter and non‐linear coupling in first order. In contrast with such solutions in most other non‐linear systems, each of these limit cycles is an exact normal mode of the unperturbed equations with no change in frequency or addition of higher harmonics.In Part I of this paper it was shown that the construction and analysis of such systems of equations are substantially simplified if the equations are expressed in terms of the normal mode co‐ordinates of the unperturbed system. the effects of the cyclic symmetry, as well as those of a higher symmetry shared by previous authors' models, were studied.It is shown here that similar results can be obtained in systems if the coupling involves a phase shift. the phase shift places added conditions on the systems, so that some sets of equations, shown to have the simple limit cycle solutions, no longer have them after shift is introduced. the methods of the earlier paper, however, can be used to find families of systems with phase shifts which have such solutions.A result in Part I, that frequencies in a system with the higher symmetry mentioned above are unchanged from those of the unperturbed system, is not valid if phase shifts are introduced.

Generalized nonlinear Doebner-Goldin Schrödinger equation and the relaxation of quantum systems

We propose a new class of nonlinear homogeneous extension of the Doebner-Goldin Schrödinger equation, valid for arbitrary representations and operators, chosen in accordance with the investigated physical problem. We verify that the nonlinearity simulates an environment, thence, the new model leads to simple exact solutions as, for instance, the time-dependent squeezed coherent states and a special class of stationary states that we call pseudothermal, reached after relaxation. We illustrate the use of the new equation with applications to problems such as, the relaxation of a two-level or spin-1/2 system, and of the harmonic oscillator (HO) or equivalently, the emission-absorption process of photons in an electromagnetic cavity. Furthermore, in order to compare solutions for the HO example we introduce two different representations in the new equation, one continuous (positional representation) and the other discrete (Fock states).

Massless fields in scalar-tensor cosmologies.

We derive exact Friedmann--Robertson--Walker cosmological solutions in general scalar--tensor gravity theories, including Brans--Dicke gravity, for stiff matter or radiation. These correspond to the long or short wavelength modes respectively of massless scalar fields. If present, the long wavelength modes of such fields would be expected to dominate the energy density of the universe at early times and thus these models provide an insight into the classical behaviour of these scalar--tensor cosmologies near an initial singularity, or bounce. The particularly simple exact solutions also provide a useful example of the possible evolution of the Brans--Dicke (or dilaton) field, $\phi$, and the Brans--Dicke parameter, $\omega(\phi)$, at late times in spatially curved as well as flat universes. We also discuss the corresponding solutions in the conformally related Einstein metric.

New families of exact solutions for finitely deformed incompressible elastic materials

The governing partial differential equations for static deformations of homogeneous isotropic incompressible elastic materials are highly nonlinear, and consequently only a few exact solutions are known. For these materials, only one general solution involving a single arbitrary function is presently known, which is for plane deformations and is applicable to the so-called Varga strain-energy function. In this paper, new families of exact solutions are derived for plane and axially symmetric deformations of perfectly elastic materials. In the case of plane deformations, a different formulation of the known general solution for the Varga strain-energy function is presented, from which numerous new solutions may be obtained. In the case of axially symmetric deformations, the Varga material again arises as the privileged strain-energy function, and this constitutes the main result of the paper. For this material, a number of new simple exact solutions are derived for axially symmetric deformations. Finally, all the solutions obtained here are shown to also apply to a modified Varga strain-energy function involving the reciprocals of the principal stretches

Some exact solutions of convection‐diffusion and diffusion equations

We examine solutions of the convection‐diffusion equation during steady radial flow in two and three dimensions with diffusivity proportional to Pn, with P the Péclet number. In two dimensions, closed‐form solutions of the Laplace‐transformed equation occur only for n = 1 and n = 2; and in three dimensions only for n = 5/4 and n = 2. The inversions for n = 1 (two dimensions) and 5/4 (three dimensions) cannot be expressed in closed form. On the other hand, the value n = 2, proposed by de Gennes (1983), leads to simple exact solutions, both in two dimensions and in three. In both cases a change of variable gives a one‐dimensional equation with constant coefficients. Exact instantaneous and continuous point source solutions for radial flow in two and three dimensions are thus established. In general, finite difference methods are needed for arbitrary n. Relevant instantaneous and continuous point source diffusion solutions are developed for radial two‐ and three‐dimensional systems. These represent small‐time solutions useful for initiating the finite difference solutions for convection‐diffusion with arbitrary n.

A Note On The Vibrations Of Infinite Elastic Plates In Contact With Water

This paper deals with the free vibrations of infinite elastic plates in the presence of water. Although this is a textbook problem, which has been studied a long time ago, no exact solution in the framework of linear theory seems to have been given so far. A very simple exact solution given in the paper permits the computation of "added virtual mass factors". It is found that the results not only differ quantitatively but also qualitatively in an important manner from the known approximate textbook solution: no threshold wavenumber for acoustic short circuit is found with the exact solution.

Numerical analysis of quasiperiodic perturbations for the Alfvén wave.

The Alfven wave may have a localized eigenfunction when it propagates on a chaotic magnetic field. The Arnold-Beltrami-Childress (ABC) flow is a paradigm of chaotic stream lines and is a simple exact solution to the three-dimensional force-free plasma equilibrium equations. The three-dimensional structure of the magnetic field is represented by sinusoidal quasiperiodic modulation. The short wavelength Alfven wave equation for the ABC-flow magnetic field has a quasiperiodic potential term, which induces interference among Bragg-reflected'' waves with irregular phases. Then the eigenfunction decays at long distance and a point spectrum occurs. Two different types of short wavelength modes have numerically analyzed to demonstrate the existence of localized Alfven wave eigenmodes.

Geodesic motion in Taub--NUT spinning space

The geodesic motion of pseudo-classical spinning particles in the Euclidean Taub--NUT space is analysed. The generalized Killing equations for spinning space are investigated and the constants of motion are derived in terms of the solutions of these equations. A simple exact solution, corresponding to trajectories lying on a cone, is given.

Exact solutions of cyclically symmetric oscillator equations with non‐linear coupling

AbstractKaplan and Yardeni have found very simple exact limit cycle solutions in cyclically symmetric systems of N oscillator equations with linear coupling in zero order of a perturbation parameter and non‐linear coupling in first order. In contrast with such solutions in other non‐linear systems, each of these limit cycles is a normal mode of the unperturbed equations, with no change in frequency. the sources of this simple behaviour are studied here with the equations expressed in terms of the normal mode co‐ordinates of the unperturbed system rather than in the original co‐ordinates. It is found that the simplicity of the Kaplan‐Yardeni solutions arises partly from an additional symmetry of the perturbation terms beyond the cyclic symmetry and partly from the specific choices of the perturbations. Extension of their systems to arbitrary N leads to the result that all such sets of equations have similar simple limit cycles. More general cyclically symmetric sets of equations are also discussed, with limit cycle solutions whose frequencies are shifted from the zero‐order values by easily calculated amounts or with solutions which are linear combinations of zero‐order normal modes.

A new class of nonlinear generalizations of the Schrodinger equation

We propose a new, nonlinear representation-independent generalized Schrodinger equation, satisfying the homogeneity condition, which is valid for arbitrary representations and arbitrary operators. This is a generalization of the equation recently proposed by Doebner and Goldin(1992) for the wavefunction written in Cartesian coordinates. The new model leads to simple exact solutions describing, for instance, the relaxation of the two-level system and the harmonic oscillator, both in Fock and in coherent states bases.

Vibrating tables — a two-dimensional network analysis

For the first time a vibrating table is modeled using network analysis. In a rectangular network of cells representing a table, particles starting at the upper left hand corner move to adjacent cells. Particles move either to their right or down with each shake of the table. A simple exact mathematical solution is found for the composition of particles in any given cell. From this model, output profiles for particle distributions exiting the table are plotted. Larger tables yield significantly sharper separations. Table tuning variables are discussed. A table, rather than being a single, simple finicky unit operation, is shown to be a sophisticated network of cells.