Special Exact Solution Research Articles

Bending of pinned dust-ion acoustic solitary waves in presence of charged space debris.

We consider a low temperature plasma environment in the low Earth orbital region in the presence of charged space debris particles. The dynamics of (2+1)-dimensional nonlinear dust-ion acoustic waves with weak transverse perturbation, generated in the system, is found to be governed by a forced Kadomtsev-Petviashvili equation, where the forcing term depends on charged space debris function. The bending phenomena of some exact dust-ion acoustic solitary wave solutions in the x-t and x-y planes are shown; they result from the consideration of different types of possible localized debris functions. A family of exact pinned accelerated solitary wave solutions has been obtained where the velocity changes over time but the amplitude remains constant. The shape of the debris function also changes during its propagation. Also, a special exact solitary wave solution has been derived for the dust-ion acoustic wave that gets curved in spatial dimensions with the curvature depending upon the nature of the forcing debris function. Such intricate solitary wave solutions may be useful in modeling real experimental data.

Chirped [formula omitted]-shaped optical solitons of Chen–Lee–Liu equation

Propagating chirped soliton solutions for the Chen–Lee–Liu equation also called the derivative nonlinear Schrödinger II equation are investigated by application of the ansatz method. The model incorporating self-steepening term has many applications in nonlinear optical fibers and plasma physics. A nonlinear differential equation describing the evolution of the wave amplitude in the nonlinear media is derived by means of the coupled amplitude-phase formulation. Special exact chirped soliton solution that takes the shape of “W” is determined for the first time in presence of all physical effects. It is shown that the nonlinear chirp associated with this type of solitons is crucially dependent on the wave intensity and related to self-steepening and group velocity dispersion parameters. Parametric conditions on system parameters for the existence of the chirped soliton structure are also presented. This soliton solution exists due to a balance among group velocity dispersion and self-steepening effect solely.

Exact Solutions Superimposed with Nonlinear Plane Waves

The flow of fluid in atmosphere and ocean is governed by rotating stratified Boussinesq equations. Through the literature, we found that many researchers are trying to find the solutions of rotating stratified Boussinesq equations. In this paper, we have obtained special exact solutions and nonlinear plane waves. Finally, we provide exact solutions to rotating stratified Boussinesq equations with large scale motion superimposed with the nonlinear plane waves. In support of our investigations, we provided two examples: one described the special exact solution and in second example, we have determined the special exact solution superimposed with nonlinear plane wave. Also, we depicted some integral curves that represent the flow of an incompressible fluid particle on the planex1+x2=L(constant)as the particular case.

A special exact spherically symmetric solution in f(T) gravity theories

A non-diagonal spherically symmetric tetrad field, involving four unknown functions of radial coordinate $r$, is applied to the equations of motion of f(T) gravity theory. A special exact vacuum solution with one constant of integration is obtained. The scalar torsion related to this special solution vanishes. To understand the physical meaning of the constant of integration we calculate the energy associated with this solution and show how it is related to the gravitational mass of the system.

Pfaffian solutions and extended Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation

Based on the Pfaffian derivative formula and Hirota bilinear method, the Pfaffian solutions to (3+1)-dimensional Jimbo–Miwa equation are obtained under a set of linear partial differential condition. Moreover, we extend the linear partial differential condition and proved that (3+1)-dimensional Jimbo–Miwa equation has extended Pfaffian solutions. As examples, special exact two-soliton solution and three-soliton solution are computed and plotted. Our results show that (3+1)-dimensional Jimbo–Miwa equation has Pfaffian solutions like BKP equation.

Thermal segregation of intruders in the Fourier state of a granular gas

A low density binary mixture of granular gases is considered within the Boltzmann kinetic theory. One component, the intruders, is taken to be dilute with respect to the other, and thermal segregation of the two species is described for a special solution to the Boltzmann equation. This solution has a macroscopic hydrodynamic representation with a constant temperature gradient and is referred to as the Fourier state. The thermal diffusion factor characterizing conditions for segregation is calculated without the usual restriction to Navier-Stokes hydrodynamics. Integral equations for the coefficients in this hydrodynamic description are calculated approximately within a Sonine polynomial expansion. Molecular dynamics simulations are reported, confirming the existence of this idealized Fourier state. Good agreement is found for the predicted and simulated thermal diffusion coefficient, while only qualitative agreement is found for the temperature ratio.

NONMINIMAL ISOTROPIC COSMOLOGICAL MODEL WITH YANG–MILLS AND HIGGS FIELDS

We establish a nonminimal Einstein–Yang–Mills–Higgs model, which contains six coupling parameters. The first three parameters relate to the nonminimal coupling of a non-Abelian gauge field and a gravity field, the next two parameters describe the so-called derivative nonminimal coupling of a scalar multiplet with a gravity field, and the sixth parameter introduces the standard coupling of a scalar field with a Ricci scalar. The formulated six-parameter nonminimal Einstein–Yang–Mills–Higgs model is applied to cosmology. We show that there exists a unique exact cosmological solution of the de Sitter type for a special choice of the coupling parameters. The nonminimally extended Yang–Mills and Higgs equations are satisfied for arbitrary gauge and scalar fields, when the coupling parameters are specifically related to the curvature constant of the isotropic space–time. Based on this special exact solution, we discuss the problem of a hidden anisotropy of the Yang–Mills field, and give an explicit example, when the nonminimal coupling effectively screens the anisotropy induced by the Yang–Mills field and thus restores the isotropy of the model.

Conformal mappings to a doubly connected polycircular arc domain

The explicit construction of the conformal mapping of a concentric annulus to a doubly connected polygonal domain was first reported by Akhiezer in 1928. The construction of an analogous formula for the case of a polycircular arc domain, i.e. for a doubly connected domain whose boundaries are a union of circular arc segments, has remained an important open problem. In this paper, we present this explicit formula. We first introduce a new method for deriving the classical formula of Akhiezer and then show how to generalize the method to the case of a doubly connected polycircular arc domain. As an analytical check of the formula, a special exact solution for a doubly connected polycircular arc mapping is derived and compared with that obtained from the more general construction. As an illustrative example, a doubly connected polycircular arc domain arising in a classic potential flow problem considered in the last century by Lord Rayleigh is considered in detail.

A note on traveling wave solutions for a class of nonlinear viscoelastic media

Traveling wave solutions (TWSs) are investigated for transverse propagation in a class of nonlinear viscoelastic media. Specifically, the propagation speed and shock thickness are determined, it is shown that vortex sheet formation occurs as the Reynolds number tends to infinity, and a special case exact solution is obtained and analyzed. Most significantly, it is proved that stable, transverse TWSs are possible only in super-Hookean media (i.e., media in which the fourth elastic modulus is positive) and that the propagation speed is a minimum when the limit at +∞ is zero.

Travelling Wave Solution of Korteweg-de Vries-Burger's Equation

An attempt has been made to obtain exact analytical travelling wave solution of Korteweg-de Vries-Burger's (KdVB) equation by the so-called tanh-method. This equation can be derived for dust ion acoustic shocks by using reduction perturbation method. It is found that an exact solution of the KdVB equation is obtained by tanh-method, provided the parameters involved satisfy a constraint relation. However a special exact analytical solution can be obtained where no such restriction is necessary. This solution has the structure of a shock wave. Numerical solution is also obtained for travelling wave with or without the assumption of the constraint relation. We have also found a singular solution in terms of cosech and coth functions.

A Validation Method for Solution of Nonlinear Differential Equations: Construction of Exact Solutions Neighboring Approximate Solutions

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed through this method have a known exact solution, even though analytical solutions are generally not obtainable. The solution is constructed such that it lies near a given approximate numerical solution, and therefore the special case solution can be generated in a versatile and physically meaningful fashion and can serve as a benchmark problem to validate approximate solution methods. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. A multi-variable orthogonal function expansion method and computer symbol manipulation are successfully used for this process. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Illustrative examples show the utility of this method not only for the ordinary differential equations (ODEs) but for the partial differential equations (PDEs).

On the Problem of the Determination of Force Distributions in Granular Heaps Using Continuum Theory

The determination of the horizontal and vertical force distributions at the base of a sand-pile is by now a famous problem in granular theory. In 1981 it was suggested from experimental work that the peak vertical force at the base does not occur directly beneath the vertex of the pile, but at some intermediate point so that there is a ring of maximum vertical pressure. Practising engineers have some reservation on this result and since that time numerous discrete theoretical and computational models of granular sand-piles have been proposed to explain this phenomenon, with varying degrees of success. For two-dimensional sand-piles, following other authors, we attempt to estimate the horizontal and vertical force distributions using the proper continuum mechanical theory of granular materials. For an infinite sand-pile we determine the force distributions at a certain height, and we argue that these forces should approximate those for a sand-pile of finite height resting on a horizontal surface. We exploit an adaptation of the classical Jenike radial flow solutions for granular flow in a converging wedge. This solution involves a stress function ψ which may be positive or negative. For both cases, numerical results indicate there is no solution satisfying all of the conditions of the stated problem, except in the special case when the angle of internal friction is ninety degrees. Except in this special case, the difficulty in determining a numerical solution might well be due to the fact that the zero surface stress conditions are generally incompatible with the yield condition. For ψ negative a formal exact parametric solution is presented for the special case of the angle of internal friction equal to ninety degrees and which coincides with an independent numerical solution. This special exact solution which is a bona fide solution of the stated problem may well be the 'only' solution of the problem, but does not predict the dip in the vertical force which is suggested from experimental work.

An exact parametric solution for granular flow in a converging wedge

The flow of granular materials in the presence of gravity through converging wedges and cones arises in many industrial situations. For both wedges and cones, and assuming an ideal cohesionless granular material which satisfies the Coulomb-Mohr yield condition, the number of simple analytical solutions is limited and generally the governing coupled ordinary differential equations need to be solved numerically. Here we show that for plane wedge flow, an exact parametric solution may be determined for the special case of the angle of internal friction \(\phi\) is assumed to be \(\pi/2\). This is the only known exact solution of these important equations involving two arbitrary constants. A general numerical solution obtained by previous authors is shown to coincide with the special exact solution.

Transformation of a dark soliton into a bright pulse

We consider a system of cubic complex Ginzburg–Landau equations governing copropagation of two waves with opposite signs of the dispersion in a nonlinear optical fiber in the presence of gain and losses. The waves are coupled by cross-phase modulation and stimulated Raman scattering. A special exact solution for a bound state of bright and dark solitons is found (unlike the well-known exact dark-soliton solution to the single complex Ginzburg–Landau equation, which is a sink, this compound soliton proves to be a source emitting traveling waves). Numerical simulations reveal that the compound soliton remains stable over ∼10 soliton periods. Next, we demonstrate that a very weak seed noise, added to an initial state in the form of a dark soliton in the normal-dispersion mode and nothing in the anomalous-dispersion one, gives rise to a process of the generation of a bright pulse in the latter mode, while the dark soliton gets grayer and eventually disappears. Thus this scheme can be used for an effective transformation of a dark soliton into a bright one, which is of interest by itself and may also find applications in photonics.

Example of shock waves in unstable media: the focusing nonlinear Schrödinger equation

Dissipationless shock waves in modulational unstable one-dimensional media are investigated for the simplest example of the integrable focusing nonlinear Schrödinger (NS) equation. Our approach is based on the construction of a special exact solution of the Whitham-NS system, which “partially saturates” the modulational instability.

Construction of Benchmark Problems for Solution of Ordinary Differential Equations

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed in this fashion have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially avail­ able approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the as­ sumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Five illus­ trative examples are presented. © 1994 John Wiley & Sons, Inc.

Construction of Benchmark Problems for Solution of Ordinary Differential Equations

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed in this fashion have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Five illustrative examples are presented.

Hilbert-class or "normal" solutions for stationary heat flow.

The concept of a Hilbert-class or ``normal'' solution in kinetic theory refers to one whose space and time dependence is determined entirely through the hydrodynamic variables. Such solutions are expected to apply sufficiently far from the boundaries. We investigate this concept quantitatively for the nonlinear Bhatnagar-Gross-Krook equation to describe a gas between two infinite parallel planes at different uniform temperatures. For sufficiently small average Knudsen numbers, spatial domains are identified for which a normal solution applies. It is shown that these conditions include states far from equilibrium, in the sense that deviations of the normal solution from the first Chapman-Enskog approximation can be large. The special exact solution of the preceding paper is found to be the normal solution in the limit of constant temperature gradient.

Thermodynamical properties of a mixed Ising ferromagnet system

The critical behavior of a mixed ferromagnetic Ising spin system consisting of spin −1 2 and spin −1, with a single-ion uniaxial crystal field included, is studied by the use of a newly developed effective-field theory. The nature of the phase diagram is analysed for square and honeycomb lattices and comparison with a special exact solution and other approximation schemes are made.

THE SPHERICALLY SYMMETRIC AND STATIC SOLUTIONS IN THE L?U(1) GAUGE THEORY OF GRAVITATION

It is more difficult to find an exact solution of the field equations in the gauge theory of gravitation as compared with finding that in Einstein's theory. As far as some problems in physics are concerned, however, it is sufficient to obtain a Newtonian approximate solution and a post-Newtonian approximate solution. In this note the L?U(1) gauge theory of gravitation is discussed. A special exact solution and the approximation to first order to the spherically symmetric and static field of a charged mass point are obtained.