General Class Of Solutions Research Articles

The Cavalcante–Tenenblat equation – Does the equation admits physical significance?

The main purpose of the given paper is to analyze a less studied third order non-linear partial differential equation, the so-called Cavalcante–Tenenblat equation (CTE) in the following form: u t = ( u x - 1 / 2 ) xx + u x 3 / 2 . Since general class of solutions are of basic interest a complete characterization of the group properties is given. The traveling wave ‘ansatz’ restricts the solution manifold to special class of solutions and hence, a generalize algorithm is necessary. We determine the Lie point symmetry vector fields and calculate similarity ‘ansätze’. Further, we also derive a few non-linear transformations and some similarity solutions are obtained explicitly. Due to the complexity of some similarity solutions a numerical procedure is of advantage. Moreover, the non-classical case (potential symmetries) is studied to the first time and further, we show how the CTE leads to approximate symmetries and we apply the method to the first time. We call the disturbed equation the CTE– ε equation and we show how to derive new class of solutions. Finally, the equation does not pass the Painlevé-test and is therefore not soluble by the Inverse Scattering Transform Method (IST). Hence, suitable alternative (algebraic) approaches are necessary to derive class of solutions explicitly.

Abelian functions associated with a cyclic tetragonal curve of genus six

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0. We construct Abelian functions using the multivariate σ-function associated with the curve, generalizing the theory of the Weierstrass ℘-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi inversion problem, a power series expansion for σ(u) and a new addition formula.

Biorthogonal Systems on the Unit Circle, Regular Semiclassical Weights, and the Discrete Garnier Equations

We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrence relations of the discrete Painlev\'e type, for example in the case with one free deformation variable the system was found to be characterised by a solution to the discrete fifth Painlev\'e equation. Here we derive the canonical forms of the multi-variable generalisation of the discrete fifth Painlev\'e equation to the Garnier systems, i.e. for arbitrary numbers of deformation variables.

Charged anisotropic matter with a linear equation of state

We consider the general situation of a compact relativistic body with anisotropic pressures in the presence of the electromagnetic field. The equation of state for the matter distribution is linear and may be applied to strange stars with quark matter. Three classes of new exact solutions are found to the Einstein–Maxwell system. This is achieved by specifying a particular form for one of the gravitational potentials and the electric field intensity. We can regain anisotropic and isotropic models from our general class of solutions. A physical analysis indicates that the charged solutions describe realistic compact spheres with anisotropic matter distribution. The equation of state is consistent with dark energy stars and charged quark matter distributions. The masses and central densities correspond to realistic stellar objects in the general case when anisotropy and charge are present.

On the theory of magneto-sound double simple waves

AbstractA two-dimensional double simple wave solution is given for both weakly and highly magnetized non-relativistic plasmas moving across the magnetic field. The dependence of the density and the magnetic field on the two independent phases, namely, components of the fluid velocity, is derived. It is shown that initial spatial distributions must satisfy a definite equation whose solution determines a special category for initial conditions. The time of blow up for any fixed value of the pair phase is found. A large general class of solutions for initial distributions is obtained. For any chosen initial distribution, the physical plane of flow at any instant of time splits into two regions, one forbidden and the other permitted. These regions are obtained numerically at a typical time for a special initial distribution. For this double wave solution, differential equations for streamlines and fluid trajectories are derived. Only for the simplest cases can the corresponding curves be completely integrated and these are given in this paper. The results are qualitatively similar to the one-dimensional case derived by Stenflo and Shukla.

Extended matter coupled to BF theory

Recently, a topological field theory of membrane matter coupled to BF theory in arbitrary spacetime dimensions was proposed [J. C. Baez and A. Perez, Adv. Theor. Math. Phys. 11, 3 (2007).]. In this paper, we discuss various aspects of the four-dimensional theory. First, we study classical solutions leading to an interpretation of the theory in terms of strings propagating on a flat spacetime. We also show that the general classical solutions of the theory are in one-to-one correspondence with solutions of Einstein's equations in the presence of distributional matter (cosmic strings). Second, we quantize the theory and present, in particular, a prescription to regularize the physical inner product of the canonical theory. We show how the resulting transition amplitudes are dual to evaluations of Feynman diagrams coupled to three-dimensional quantum gravity. Finally, we remove the regulator by proving the topological invariance of the transition amplitudes.

A note on new solitary and similarity class of solutions of a fourth-order non-linear evolution equation

In this paper, Lie’s method is used to calculate new class of solutions of a less studied fourth-order non-linear partial differential equation. In our previous paper [A. Huber, Appl. Math. Comp. 166/2 (2005) 464], we have applied the tanh-method to generate solutions. In this case, special class of solutions in form of traveling waves result (single soliton solutions as well as class of irregular solutions). Therefore, general families of solutions are of basic interest. Doing so, new results can be found in [A. Huber, Appl. Math. Comp., in press] using the Weierstraß transformation derived first by the author. Moreover, a complete characterization of the group properties is given because group properties are unknown up to now. We determine the Lie point symmetry vector fields and calculate similarity “ansätze”. Further, we also derive a few non-linear transformations and some similarity solutions are obtained explicitly. The main purpose for the application of Lie’s method is of course the fact that we are able to calculate class of general solutions which do not underlie such strong restrictions as in the case of traveling wave “ansätze”. Otherwise, it is necessary to perform a group analysis in order to improve the solution manifold by an alternative way. In addition, the studied equation passes the Painlevé-test and seems to be integrable. General class of solutions in terms of elliptic functions are derived via Lie’s approach for the first time representing exact solitary wave propagation and surprisingly, cusp–solitary class of solutions are obtained.

Cosmological models in scalar tensor theories of gravity and observations: a class of general solutions

Aims. We study cosmological models in scalar tensor theories of gravity with power-law potentials as models of an accelerating universe. Methods. We consider cosmological models in scalar tensor theories of gravity that describe an accelerating universe and study a family of inverse power-law potentials, for which exact solutions of the Einstein equations are known. We also compare theoretical predictions of our models with observations. For this we use the following data: the publicly available catalogs of type Ia supernovae and high redshift gamma ray bursts, the parameters of large-scale structure determined by the 2-degree Field Galaxy Redshift Survey (2dFGRS), and measurements of cosmological distances based on the Sunyaev-Zel’dovich effect, among others. Results. We present a class of cosmological models that describe the evolution of a homogeneous and isotropic universe filled with dust-like matter and a scalar field that is non minimally-coupled to gravity. We show that this class of models depends on three parameters: V0 – the amplitude of the scalar field potential, � H0 – the present value of the Hubble constant, and a real parameter s that determines the overall evolution of the universe. It turns out that these models have a very interesting feature naturally producing an epoch of accelerated expansion. We fix the values of these parameters by comparing predictions of our model with observational data. It turns out that our model is compatible with the presently available observational data.

Properties of some conformal field theories with M-theory duals

By studying classes of supersymmetric solutions of D=11 supergravity with AdS_5 factors, we determine some properties of the dual four-dimensional N=1 superconformal field theories. For some explicit solutions we calculate the central charges and also the conformal dimensions of certain chiral primary operators arising from wrapped membranes. For the most general class of solutions we show that there is a consistent Kaluza-Klein truncation to minimal D=5 gauged supergravity. This latter result allows us to study some aspects of the dual strongly coupled thermal plasma with a non-zero R-charge chemical potential and, in particular, we show that the ratio of the shear viscosity to the entropy density has the universal value of 1/4 pi.

The index of the overlap Dirac operator on a discretized 2d non-commutative torus

The index, which is given in terms of the number of zero modes of the Dirac operator with definite chirality, plays a central role in various topological aspects of gauge theories. We investigate its properties in non-commutative geometry. As a simple example, we consider the U(1) gauge theory on a discretized 2d non-commutative torus, in which general classical solutions are known. For such backgrounds we calculate the index of the overlap Dirac operator satisfying the Ginsparg-Wilson relation. When the action is small, the topological charge defined by a naive discretization takes approximately integer values, and it agrees with the index as suggested by the index theorem. Under the same condition, the value of the index turns out to be a multiple of N, the size of the 2d lattice. By interpolating the classical solutions, we construct explicit configurations, for which the index is of order 1, but the action becomes of order N. Our results suggest that the probability of obtaining a non-zero index vanishes in the continuum limit, unlike the corresponding results in the commutative space.

A note on class of traveling wave solutions of a non-linear third order system generated by Lie’s approach

Abstract In this paper, Lie’s method is used to calculate solutions of a third order non-linear system of partial differential equations (nPDE). In our previous paper [Huber A. Appl Math Comput 2005;166/2:464], we have applied the tanh-method to generate solutions, in this case special class of solutions in form of traveling wave results (single soliton solutions as well as class of irregular solutions). Therefore, general families of solutions are of basic interest. Moreover, a complete characterization of the group properties is given. We determine the Lie point symmetry vector fields and calculate similarity “ansatze” for the first time. Further, we also derive a few non-linear transformations and some similarity solutions are obtained. The main purpose for the application of Lie’s method is of course the fact that we are able to calculate class of general solutions which do not underlie such strong restrictions as in the case of traveling wave “ansatze”. Otherwise, it is necessary to perform a group analysis in order to improve the solution manifold by an alternative way. Moreover, a criterion for the integrability via the Painleve-conjecture is given and further, families of solutions in term of elliptic functions are derived via Lie‘s approach for the first time. Although no extensive studies are known up to this time a physical background of the considered system cannot exclude in future.

Relationship between (2 + 1) and (3 + 1)-Friedmann–Robertson–Walker cosmologies; linear, non-linear, and polytropic state equations

It is shown that Friedmann–Robertson–Walker (FRW) cosmological models coupled to a single scalar field and to a perfect fluid fitting a wide class of matter perfect fluid state equations, determined in (3+1) dimensional gravity can be related to their (2+1) cosmological counterparts, and vice-versa, by using simple algebraic rules relating gravitational constants, state parameters, perfect fluid and scalar field characteristics. It should be pointed out that the demonstration of these relations for the scalar fields and potentials does not require the fulfilment of any state equation for the scalar field energy density and pressure. As far as to the perfect fluid is concerned, one has to demand the fulfilment of state equations of the form p+ρ = γ f(ρ). If the considered cosmologies contain the inflation field alone, then any (3+1) scalar field cosmology possesses a (2+1) counterpart, and vice-versa. Various families of solutions are derived, and we exhibited their correspondence; for instance, solutions for pure matter perfect fluids and single scalar field fulfilling linear state equations, solutions for scalar fields coupled to matter perfect fluids, a general class of solutions for scalar fields subjected to a state equation of the form p φ+ρφ = γρφ β are reported, in particular Barrow–Saich, and Barrow–Burd–Lancaster–Madsen solutions are exhibited explicitly, and finally perfect fluid solutions for polytropic state equations are given.

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.

Cosmological solutions in multidimensional model with multiple exponential potential

A family of cosmological solutions with $(n+1)$ Ricci-flat spaces in the theory with several scalar fields and multiple exponential potential is obtained when coupling vectors in exponents obey certain relations. Two subclasses of solutions with power-law and exponential behaviour of scale factors are singled out. It is proved that power-law solutions may take place only when coupling vectors are linearly independent and exponential dependence occurs for linearly dependent set of coupling vectors. A subfamily of solutions with accelerated expansion is singled out. A generalized isotropization behaviours of certain classes of general solutions are found. In quantum case exact solutions to Wheeler-DeWitt equation are obtained and special "ground state" wave functions are considered.

EXACT SOLUTIONS AND THE COSMOLOGICAL CONSTANT PROBLEM IN DILATONIC-DOMAIN-WALL HIGHER-CURVATURE STRING GRAVITY

In this paper, we extend the previous work by the authors, and elaborate further on the structure of the general solution to the graviton and dilaton equations of motion in brane world scenaria, in the context of five-dimensional effective actions with [Formula: see text] higher-curvature corrections, compatible with bulk string-amplitude calculations. We consider (multi)brane scenaria, dividing the bulk space into regions, in which one matches two classes of general solutions, a linear Randall–Sundrum solution and a (logarithmic) dilatonic domain wall (bulk naked singularity). We pay particular attention to examining the possibility of resolving the mass hierarchy problem together with the vanishing of the vacuum energy on the observable world, which is taken to be a positive tension brane. The appearance of naked dilatonic domain walls provides a dynamical restriction of the bulk space–time. Of particular interest is a dilatonic-wall solution, which after appropriate coordinate transformation, results in a linear dilaton conformal field theory. The latter may provide a holographic resolution of the naked singularity problem. All the string-inspired models involved have the generic feature that the brane tensions are proportional to the string coupling gs; it remains a challenge for string theory, therefore, to show whether microscopic models respecting this feature can be constructed.

General classical solutions in the noncommutative CPN−1 model

We give an explicit construction of general classical solutions for the noncommutative CPN−1 model in two dimensions, showing that they correspond to integer values for the action and topological charge. We also give explicit solutions for the Dirac equation in the background of these general solutions and show that the index theorem is satisfied.

Systematic study of the radion in the compact Randall-Sundrum model

We systematically study the question of identification and consistent inclusion of the radion, within the Lagrangian approach, in a two-brane Randall-Sundrum model. Exploiting the symmetry properties of the theory, we show how the radion can be identified unambiguously and give the action to all orders in the radion field and the metric. Using the background field method, we expand the theory to quadratic orders in the fields. We show that the most general classical solutions, for the induced metric on the branes in the case of a constant radion and a factorizable four-dimensional metric, correspond to Einstein spaces. We discuss extensively the diagonalization of the quadratic action. Furthermore, we obtain the four-dimensional effective theory from this and study the question of the spectrum as well as the couplings in these theories.

On a general class of wormhole geometries

A general class of solutions is obtained which describe aspherically symmetric wormhole system. The presence of arbitraryfunctions allows one to describe infinitely many wormhole systems ofthis type. The source of the stress-energy supporting the structureconsists of an anisotropic brown dwarf `star' which smoothly joins thevacuum and may possess an arbitrary cosmological constant. It isdemonstrated how this set of solutions allows for a non-zero energydensity and therefore allows positive stellar mass as well as howviolations of energy conditions may be minimized. Unlike examplesconsidered thus far, emphasis here is placed on construction bymanipulating the matter field as opposed to the metric. This scheme isgenerally more physical than the purely geometric method. Finally,explicit examples are constructed including an example whichdemonstrates how multiple closed universes may be connected by suchwormholes. The number of connected universes may be finite or infinite.

The diophantine approximation of general solutions of algebraic differential equations

Hoping to find bounds on the approximation by rational functions of all members of certain classes of power series, it is necessary to restrict to “reasonable” classes of functions. Algebraic functions are one such class. Another class is the set of irrational power series solutions of linear homogeneous differential equations with polynomial coefficients. Tight bounds have been found in both cases. The jump to nonlinear differential equations is another matter. The present paper obtains tight bounds, if one restricts to certain classes of general solutions. The presence of enough algebraically independent parameters permits the desired type of bound. It is to be hoped that the more general case will ultimately be settled; however, other methods will certainly be needed.

General classical solution for dynamics of charges with radiation reaction

The relativistic dynamics of a charge in the presence of the radiation reaction force is solved in general. It is shown that the solution admits mass conversion of the charge into the energy of the electromagnetic field. The mechanism of annihilation is demonstrated on the example of the radiative capture of the electron by the proton.