Family Of Analytical Solutions Research Articles

A similarity reduction of the generalized Grad–Shafranov equation

We extend previous work [Y. E. Litvinenko, Phys. Plasmas 17, 074502 (2010)] on a direct method for finding similarity reductions of partial differential equations such as the Grad–Shafranov equation, to the case of the generalized Grad–Shafranov equation (GGSE) with arbitrary incompressible flow. Several families of analytic solutions are constructed, the generalized Solovév solution being a particular case, which contain both the classical and non-classical group-invariant solutions to the GGSE. Those solutions can describe a variety of equilibrium configurations pertinent to toroidal magnetically confined plasmas and planetary magnetospheres.

New abundant analytic solutions for generalized KdV6 equation with time-dependent variable coefficients using Painlevé analysis and auto-Bäcklund transformation

This paper considers the generalized KdV6 equation with time-dependent variable coefficients. The integrability of the considered equation is being examined by the Painlevé analysis method. Further, an auto-Bäcklund transformation method has been adopted to obtain the analytic solutions. Using this technique, five novel families of analytic solutions in the form of rational, exponential, hyperbolic and trigonometric functions have been successfully found for the considered equation. New kink-antikink and periodic soliton solutions have been discovered using this method. The solutions are graphically depicted to show the physical significance of the problem under consideration.

Topological or rotational non-Abelian gauge fields from Einstein-Skyrme holography

We report analytically known states at non-zero temperature which may serve as a powerful tool to reveal common topological and thermodynamic properties of systems ranging from the QCD phase diagram to topological phase transitions in condensed matter materials. In the holographically dual gravity theory, these are analytic solutions to a five-dimensional non-linear-sigma (Skyrme) model dynamically coupled to Einstein gravity. This theory is shown to be holographically dual to mathcal{N} = 4 Super-Yang-Mills theory coupled to an SU(2)-current. All solutions are fully backreacted asymptotically Anti-de Sitter (AdS) black branes or holes. One family of global AdS black hole solutions contains non-Abelian gauge field configurations with positive integer Chern numbers and finite energy density. Larger Chern numbers increase the Hawking-Page transition temperature. In the holographically dual field theory this indicates a significant effect on the deconfinement phase transition. Black holes with one Hawking temperature can have distinct Chern numbers, potentially enabling topological transitions. A second family of analytic solutions, rotating black branes, is also provided. These rotating solutions induce states with propagating charge density waves in the dual field theory. We compute the Hawking temperature, entropy density, angular velocity and free energy for these black holes/branes. These correspond to thermodynamic data in the dual field theory. For these states the energy-momentum tensor, (non-)conserved current, and topological charge are interpreted.

An approach to exact nonlinear feedback control design employing recursive approximations and the null space

A strategy to design exact nonlinear feedback controllers based on a recursive application of approximate linearization methods is examined. The computations are algebraic and computationally simpler than solving the set of coupled nonlinear partial differential equations thereby facilitating practical symbolic computer computations enabling discernment of evolving patterns in the approximate solutions as the order of approximation increases. Utilizing the null space that appears at each step in the computations as part of the computations, a family of analytic solutions can be generated asymptotically. There are possibilities for optimizing the performance by judiciously choice of analytic solution that emerge from the selective use of the null space.

Explicit solutions with non-trivial phase of the inhomogeneous coupled two-component NLS system

In this article, we construct novel explicit solutions for nonlinear Schrödinger systems with spatially inhomogeneous nonlinearity by means of the Lie symmetry method. We focus the attention to solutions with the non-trivial phase, which have been scarcely considered in the related literature. To get started, the theoretical method based on Lie symmetries is exposed, thus reducing the problem to the integrability of an ordinary differential equation (ODE). The non-trivial phase introduces a singular term into the ODE. Then, the method is used to construct new families of analytical solutions. Some illustrative examples are provided.

Families of analytic solutions for (2+1) model in unbounded domain via optimal Lie vectors with integrating factors

Through the commutator table and the adjoint table between the infinitesimals, we apply two stages of Lie symmetry reduction to reduce the (2[Formula: see text]+[Formula: see text]1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation to ordinary differential equations (ODE’s). Some of these ODE’s had no quadrature. We derive several new solutions for these non-solvable ODE’s using Integrating Factors property.

(Compactified) black branes in four dimensional f(R)-gravity

A new family of analytical solutions in a four dimensional static spacetime is presented for f(R)-gravity. In contrast to General Relativity, we find that a non trivial black brane/string solution is supported in vacuum power law f(R)-gravity for appropriate values of the parameters characterizing the model and when axisymmetry is introduced in the line element. For the aforementioned solution, we perform a brief investigation over its basic thermodynamic quantities.

Manifolds of solutions for Hirzebruch functional equations

For the nth Hirzebruch equation we introduce the notion of universal manifold Mn of formal solutions. It is shown that the manifold Mn, where n > 1, is algebraic and its dimension is not greater than n + 1. We give a family of polynomials generating the relation ideal in the polynomial ring on Mn. In the case n = 2 the generators of this ideal are described. As a corollary we obtain an effective description of the manifold M2 and therefore all series determining complex Hirzebruch genera that are fiberwise multiplicative on projectivizations of complex vector bundles. A family of analytic solutions of the second Hirzebruch equation is described in terms of Weierstrass elliptic functions and in terms of Baker–Akhiezer functions of elliptic curves. For this functions the curves differ, yet the series expansions in the vicinity of 0 coincide.

A refined holographic QCD model and QCD phase structure

We consider the Einstein-Maxwell-dilaton system with an arbitrary kinetic gauge function and a dilaton potential. A family of analytic solutions is obtained by the potential reconstruction method. We then study its holographic dual QCD model. The kinetic gauge function can be fixed by requesting the linear Regge spectrum of mesons. We calculate the free energy to obtain the phase diagram of the holographic QCD model.

Phase structure in a dynamical soft-wall holographic QCD model

We consider the Einstein-Maxwell-dilaton system with an arbitrary kinetic gauge function and a dilaton potential. A family of analytic solutions is obtained by the potential reconstruction method. We then study its holographic dual QCD model. The kinetic gauge function can be fixed by requesting the linear Regge spectrum of mesons. We calculate the free energy to obtain the phase diagram of the holographic QCD model and interpret our result as the heavy quarks system by comparing the recent lattice QCD simulation. We finally obtain the equations of state in our model.

Localized Nonlinear Waves in Nonlinear Schr¨odinger Equation with Nonlinearities Modulated in Space and Time

In this paper, the generalized sub-equation method is extended to investigate localized nonlinear waves of the one-dimensional nonlinear Schrödinger equation (NLSE) with potentials and nonlinearities depending on time and on spatial coordinates. With the help of symbolic computation, three families of analytical solutions of this NLS-type equation are presented. Based on these solutions, periodically and quasiperiodically oscillating solitons (dark and bright) and moving solitons are observed. Some implications to Bose-Einstein condensates are also discussed

Stability of solitons in parity-time-symmetric couplers

Families of analytical solutions are found for symmetric and antisymmetric solitons in a dual-core system with Kerr nonlinearity and parity-time (PT)-balanced gain and loss. The crucial issue is stability of the solitons. A stability region is obtained in an analytical form, and verified by simulations, for the PT-symmetric solitons. For the antisymmetric ones, the stability border is found in a numerical form. Moving solitons of both types collide elastically. The two soliton species merge into one in the "supersymmetric" case, with equal coefficients of gain, loss, and intercore coupling. These solitons feature a subexponential instability, which may be suppressed by periodic switching ("management").

Blow-up of smooth solutions to the Navier-Stokes-Poisson equations

The main purpose of this paper is concerned with blow-up smooth solutions to Navier–Stokes–Poisson (N-S-P) equations. First, we present a sufficient condition on the blow up of smooth solutions to the N-S-P system. Then we construct a family of analytical solutions that blow up in finite time. Copyright © 2010 John Wiley & Sons, Ltd.

ANALYTICAL CONSIDERATIONS ABOUT THE COSMOLOGICAL CONSTANT AND DARK ENERGY

The accelerated expansion of the universe has now been confirmed by several independent observations including those of high redshift type Ia supernovae, and the cosmic microwave background combined with the large scale structure of the universe. Another way of presenting this kinematic property of the universe is to postulate the existence of a new and exotic entity, with negative pressure, the dark energy (DE). In spite of observationally well established, no single theoretical model provides an entirely compelling framework within which cosmic acceleration or DE can be understood. At present all existing observational data are in agreement with the simplest possibility that the cosmological constant be a candidate for DE. This case is internally self-consistent and noncontradictory. The extreme smallness of the cosmological constant expressed in either Planck, or even atomic units means only that its origin is not related to strong, electromagnetic, and weak interactions. Although in this case DE reduces to only a single fundamental constant we still have no derivation from any underlying quantum field theory for its small value. From the principles of quantum cosmologies, for example, it is possible to obtain the reason for an inverse-square law for the cosmological constant with no conflict with observations. Despite the fact that this general expression is well known, in this work we introduce families of analytical solutions for the scale factor different from the current literature. The knowledge of the scale factor behavior might shed some light on these questions mentioned above since the entire evolution of a homogeneous isotropic universe is contained in the scale factor. We use different parameters for these solutions and with these parameters we establish a connection with the equation of state for different DE scenarios.

Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation

There has been considerable interest in the study on the variable-coefficient nonlinear evolution equations in recent years, since they can describe the real situations in many fields of physical and engineering sciences. In this paper, a generalized variable-coefficient KdV (GvcKdV) equation with the external-force and perturbed/dissipative terms is investigated, which can describe the various real situations, including large-amplitude internal waves, blood vessels, Bose-Einstein condensates, rods and positons. The Painleve analysis leads to the explicit constraint on the variable coefficients for such a equation to pass the Painleve test. An auto-Backlund transformation is provided by use of the truncated Painleve expansion and symbolic computation. Via the given auto-Backlund transformation, three families of analytic solutions are obtained, including the solitonic and periodic solutions.

Robertson-Walker Fluid Sources Endowed with Rotation Characterised by Quadratic Terms in Angular Velocity Parameter

Einstein's equations for a Robertson-Walker fluid source endowed with rotation are presented up to and including quadratic terms in angular velocity parameter. A family of analytic solutions are obtained for the case in which the source angular velocity is purely time-dependent. A subclass of solutions is presented which merge smoothly to homogeneous rotating and non-rotating central sources. The particular solution for dust endowed with rotation is presented. In all cases explicit expressions, depending sinusoidally on polar angle, are given for the density and internal supporting pressure of the rotating source. In addition to the non-zero axial velocity of the fluid particles it is shown that there is also a radial component of velocity which vanishes only at the poles. The velocity four-vector has a zero component between poles.

New black holes in the brane world?

It is known that the Einstein field equations in five dimensions admit more general spherically symmetric black holes on the brane than four-dimensional general relativity. We propose two families of analytic solutions (with g_tt\not=-1/g_rr), parameterized by the ADM mass and the PPN parameter beta, which reduce to Schwarzschild for beta=1. Agreement with observations requires |\beta-1| |\eta|<<1. The sign of eta plays a key role in the global causal structure, separating metrics which behave like Schwarzschild (eta<0) from those similar to Reissner-Nordstroem (eta>0). In the latter case, we find a family of black hole space-times completely regular.

The Structure of Solutions of an Elliptic Equation with a Semiregular Degeneracy

We study an elliptic equation with semiregular degeneracy on a hyperplane and obtain sufficient conditions under which the equation has one or two families of analytic solutions in a neighborhood of the variety of degeneracy. We present expressions of solutions in the form of power series.

EXACT SOLUTIONS FOR SELF-DUAL SU(2) GAUGE THEORY WITH AXIAL SYMMETRY

A model of SU(2) gauge theory is constructed in terms of local gauge-invariant variables defined over a four-dimensional space–time endowed with axial symmetry. A metric tensor gμν is defined starting with the components [Formula: see text] of the strength tensor and its dual [Formula: see text]. The components gμν are interpreted as new local gauge-invariant variables. Imposing the condition that the new metric coincides with the initial metric we obtain the field equations for the considered ansatz. We obtain the same field equations using the condition of self-duality. It is concluded that the self-dual variables are compatible with the axial symmetry of the space–time. A family of analytical solutions of the gauge field equations is also obtained. The solutions have the confining properties. All the calculations are performed using the GRTensorII computer algebra package, running on the MapleV platform.

Equilibrium and Stability of Magnetostatic Atmospheres. I. Dungey‐Type Isothermal States

Two families of analytical solutions are presented that describe isothermal magnetostatic atmospheres in uniform gravity and varying in two Cartesian dimensions. The solutions in each family share a common set of magnetic flux surfaces but have different profiles of field intensity, magnetic shear, and plasma distribution across these flux surfaces such that force balance is satisfied. A family of solutions with this mathematical degree of freedom had previously been found by Dungey. The construction and properties of the new solutions are described. The hydromagnetic stability of these solutions is discussed by using the sufficiency criteria of Hu to determine those profiles of magnetic field and plasma that assure stability for the whole system. Among the stable equilibria found are examples of sheared magnetic structures intruding into a uniformly magnetized, isothermal atmosphere. One of the two families of solutions is extended to equilibrium states that vary fully with all three Cartesian coordinates. These extended solutions allow the possibility of constructing complex structures by juxtapositioning discrete three-dimensional magnetic structures built separately. These two-dimensional and three-dimensional magnetostatic states may be useful as initial states for numerical simulation of time-dependent magnetohydrodynamic processes in the solar atmosphere.