Subclass Of Solutions Research Articles

Phase space renormalization and finite BMS charges in six dimensions

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}+ — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively.

On stable exponential cosmological solutions with two factor spaces in (1+ m + 2)-dimensional Einstein-Gauss-Bonnet model with Λ-term.

A [Formula: see text]-dimensional Einstein-Gauss-Bonnet gravitational model including the Gauss-Bonnet term and the cosmological term [Formula: see text] is considered. Exact solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters [Formula: see text] and [Formula: see text], corresponding to factor spaces of dimensions [Formula: see text] and [Formula: see text], respectively, are found. Under certain restrictions on [Formula: see text], the stability of the solutions in a class of cosmological solutions with diagonal metrics is proved. A subclass of solutions with small enough variation of the effective gravitational constant [Formula: see text] is considered and the stability of all solutions from this subclass is shown. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

On a general class of solutions for an optimal velocity model on an infinite lane

We consider a general class of car-following models. Such models were studied over the last decades both on a circular and on an infinite road. Many interesting phenomena have been studied, e.g. loss of stability and stop and go waves. Whereas there is good mathematical understanding for the circular road, for the infinite road only partial results are available. In this paper we study a general approach to obtain solutions which are valid for the different settings mentioned above. The well known solutions on the circular road can be seen as a subclass of solutions, the known approaches on the infinite lane can be identified and new solutions on the infinite lane are studied. Beside the classical tools from linear analysis we will employ nonlinear bifurcation techniques. Throughout the paper as a prominent example we will use the so-called optimal velocity model proposed in Bando, et al. [1995. “Dynamical model of traffic congestion and numerical simulation.” Physical Review E 51 (2): 1035–1042].

A Introduction to the Classical Spiral Electrodynamics: The” Spiral-Spin”

This paper demonstrates the existence of analytical solutions of the Lorentz equation for charged particles in “uniform pilot time-varying magnetic fields". These analytical solutions represent a temporal generalization of the Larmor's orbits and are expressed through a Schwarz-Christoffel spiral mapping or in spiral coordinates.
 The concepts of "spiral-spin” moment and "polar-spiral" angular momentum are then presented, the existence of a subclass of solutions for which these two angular moments are conserved is demonstrated.
 It is also shown that under the action of the "pilot fields," there exist particular trajectories for which the charged particles have a "spiral-spin" momentum constant proportional to +1/2 (solution named "spiral-spin-up ") and -1/2 (solution named "spiral-spin-down "), respectively.
 The results are in full agreement with the ideas of L.DeBroglie and A. Einstein on the possible existence of pilot fields able to describe the physical reality deterministically.
 Finally, the solution of the Lorentz equation is discussed with the WKB (Wentzel-Kramers-Brillouin) method for a superposition of two uniform magnetic fields with the same direction, the first constant and the second time-varying.

Exponential Cosmological Solutions with Three Different Hubble-Like Parameters in (1 + 3 + k1 + k2)-Dimensional EGB Model with a Λ-Term

A D-dimensional Einstein–Gauss–Bonnet model with a cosmological term Λ , governed by two non-zero constants: α 1 and α 2 , is considered. By restricting the metrics to diagonal ones, we study a class of solutions with the exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H > 0 , h 1 , and h 2 , obeying 3 H + k 1 h 1 + k 2 h 2 ≠ 0 and corresponding to factor spaces of dimensions: 3, k 1 > 1 , and k 2 > 1 , respectively, with D = 4 + k 1 + k 2 . The internal flat factor spaces of dimensions k 1 and k 2 have non-trivial symmetry groups, which depend on the number of compactified dimensions. Two cases: (i) 3 < k 1 < k 2 and (ii) 1 < k 1 = k 2 = k , k ≠ 3 , are analyzed. It is shown that in both cases, the solutions exist if α = α 2 / α 1 > 0 and α Λ > 0 obey certain restrictions, e.g., upper and lower bounds. In Case (ii), explicit relations for exact solutions are found. In both cases, the subclasses of stable and non-stable solutions are singled out. Case (i) contains a subclass of solutions describing an exponential expansion of 3 d subspace with Hubble parameter H > 0 and zero variation of the effective gravitational constant G.

On Hiemenz flow of Maxwell incompressible viscoelastic medium

Steady axisymmetric flow of a viscoelastic incompressible fluid near the critical point of the plane surface is considered. Model of Maxwellian fluid with upper convected derivative in the rheological constitutive law is used. Velocity and extra stresses field are discussed for various Weissenberg numbers. The subclass of solutions in a stationary case is completely described. The asymptotic solutions for small Weissenberg numbers are compared to the solutions of the complete nonlinear problem. It is found that the fluid elasticity decreases the boundary layer thickness.

Exact exponential cosmological solutions with two factor spaces of dimension m in EGB model with a Λ-term

A [Formula: see text]-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological term [Formula: see text] is considered. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters [Formula: see text] and [Formula: see text], corresponding to two factor spaces of dimension [Formula: see text] and obeying [Formula: see text]. We prove that the solutions, obeying [Formula: see text], are stable (in a class of cosmological solutions with diagonal metrics) for [Formula: see text] and they are unstable for [Formula: see text]. A subclass of solutions with small enough variation of the effective gravitational constant [Formula: see text] is considered. It is shown that all solutions from this subclass are stable.

On a Subclass of Solutions of the 2D Navier–Stokes Equations with Constant Energy and Enstrophy

In this paper, we are interested in studying the existence of the nonstationary solutions in the global attractor of the 2D Navier–Stokes equations with constant energy and enstrophy. We particularly focus on a subclass of these solutions that the geometric structures have a supplementary stability property which were called “chained ghost solutions” in Tian and Zhang (Indiana Univ Math J 64:1925–1958, 2015). By solving a Galerkin system of a particular form, we show the nonexistence of the chained ghost solutions for certain cases.

Stable exponential cosmological solutions with two factor spaces in the Einstein–Gauss–Bonnet model with a $$\Lambda $$ Λ -term

We study $D$-dimensional Einstein-Gauss-Bonnet gravitational model including the Gauss-Bonnet term and the cosmological term $\Lambda$. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters $H >0$ and $h$, corresponding to factor spaces of dimensions $m >2$ and $l > 2$, respectively. These solutions contain a fine-tuned $\Lambda = \Lambda (x, m, l, \alpha)$, which depends upon the ratio $h/H = x$, dimensions of factor spaces $m$ and $l$, and the ratio $\alpha = \alpha_2/\alpha_1$ of two constants ($\alpha_2$ and $\alpha_1$) of the model. The master equation $\Lambda(x, m, l,\alpha) = \Lambda$ is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. The explicit solution for $m = l$ is presented in Appendix. Imposing certain restrictions on $x$, we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. We also consider a subclass of solutions with small enough variation of the effective gravitational constant $G$ and show the stability of all solutions from this subclass.

Stable exponential cosmological solutions with 3- and l-dimensional factor spaces in the Einstein\u2013Gauss\u2013Bonnet model with a Lambda -term

A D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term Lambda is studied. We assume the metrics to be diagonal cosmological ones. For certain fine-tuned Lambda , we find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters H >0 and h, corresponding to factor spaces of dimensions 3 and l > 2, respectively and D = 1 + 3 + l. The fine-tuned Lambda = Lambda (x, l, alpha ) depends upon the ratio h/H = x, l and the ratio alpha = alpha _2/alpha _1 of two constants (alpha _2 and alpha _1) of the model. For fixed Lambda , alpha and l > 2 the equation Lambda (x,l,alpha ) = Lambda is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals (the example l =3 is presented). For certain restrictions on x we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. A subclass of solutions with small enough variation of the effective gravitational constant G is considered. It is shown that all solutions from this subclass are stable.

Boundary regularity for a parabolic obstacle type problem

We study the regularity of the free boundary, near contact points with the fixed boundary, for a parabolic free boundary problem \begin{align*} &Δu–∂u/∂t = χ_{\{u≠0\}} && \text{in } Q_r^+ = \{(x, t) \in B_r \times (–r^2, 0); \ x_1 > 0\}, \\ &u = f(x, t) &&\text{on } \{x_1 = 0\} ∩ Q_r. \end{align*} We will show that under certain regularity assumptions on the boundary data f the free boundary is a C^1 manifold up to the fixed boundary. We also show that the C^1 modulus of continuity is uniform for a certain, and specified, subclass of solutions.

Stationary and axisymmetric solutions of higher-dimensional general relativity

We study stationary and axisymmetric solutions of General Relativity, i.e. pure gravity, in four or higher dimensions. D-dimensional stationary and axisymmetric solutions are defined as having D-2 commuting Killing vector fields. We derive a canonical form of the metric for such solutions that effectively reduces the Einstein equations to a differential equation on an axisymmetric D-2 by D-2 matrix field living in three-dimensional flat space (apart from a subclass of solutions that instead reduce to a set of equations on a D-2 by D-2 matrix field living in two-dimensional flat space). This generalizes the Papapetrou form of the metric for stationary and axisymmetric solutions in four dimensions, and furthermore generalizes the work on Weyl solutions in four and higher dimensions. We analyze then the sources for the solutions, which are in the form of thin rods along a line in the three-dimensional flat space that the matrix field can be seen to live in. As examples of stationary and axisymmetric solutions, we study the five-dimensional rotating black hole and the rotating black ring, write the metrics in the canonical form and analyze the structure of the rods for each solution.

Gravitation in the Fractal D = 2 Inertial Universe: New Phenomenology in Spiral Discs and a Theoretical Basis for MOND

A particular interpretation of Mach's Principle led us to ask if it was possible to have a globally inertial universe that was irreducibly associated with a non-trivial global matter distribution, Roscoe [1]. This question received a positive answer, subject to the condition that the global matter distribution is necessarily fractal, D = 2. The present paper shows how gravitational processes can arise in this universe. We begin by showing how classical Newtonian gravitation arises from point-source perturbations of this D = 2 inertial background. We then use the insights gained from this initial analysis to arrive at a general theory for arbitrary material distributions. We illustrate the process by using it to model an idealized spiral galaxy. One particular subclass of solutions, (the logarithmic spiral) has already been extensively tested (Roscoe [2, 3]), and shown to resolve large samples of optical rotation curve data to a very high statistical precision. These analyses also led to the discovery of a major new phenomenology in spiral discs—that of discrete dynamical classes, [3]. In this paper, we analyse the theory more comprehensively, showing how this phenomenology has a possible explanation in terms of an algebraic consistency condition which must necessarily be satisfied. Of equal significance, we apply the theory with complete success to the detailed modelling of eight Low Surface Brightness spirals (LSBs) which, hitherto, have been successfully modelled only by the MOND algorithm (Milgrom [5–7]. We are able to conclude that the essence of the MOND algorithm must be contained within the presented theory.

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.

Electromagnetic waves in a rigidly rotating frame

Vacuum solutions of Maxwell equations in a rigidly rotating frame are derived in order to specify how the rotating observer would measure passing electromagnetic radiation. Approximate analytic solutions are found for the regions close to the light cylinder and the rotation axis, respectively. It is found that despite the singularity of the reference frame at the light cylinder, there is a subclass of solutions, which is regular at this distance. The results and their possible astrophysical applications are discussed.

Two-Fluid Bianchi Type II Cosmological Models

In this paper we present a class of solutions of Einstein's field equations describing two-fluid models of the universe in a locally rotationally symmetric Bianchi type II space-time. In these models one fluid is the radiation distribution which represents the cosmic microwave background and the other fluid is the perfect fluid representing the matter content of the universe. It is found that both the fluids are comoving in the locally rotationally symmetric Bianchi type II space-time. The behaviour of the radiation density, matter density, the ratio of the matter density to the radiation density and the pressure has been discussed. A subclass of solutions is found to describe models of a spatially homogeneous and partially isotropic universe evolving from a radiation dominated era to a pressure free matter dominated era.

Non-resonant interacting water waves in 2+1 dimensions

We investigate the interaction among small amplitude water waves, when the fluid motion is in a basin of arbitrary, uniform depth. Waves are supposed to be non-resonant, i.e., with different group velocities that are not close to each other. Starting from the isotropic pseudo-differential Milewski–Keller equation and using an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling, we show that the amplitude slow modulation of Fourier modes can be described by a model system of non-linear evolution equations. We demonstrate that the system is C-integrable, i.e., can be linearized through an appropriate transformation of the dependent and independent variables. A subclass of solutions gives rise to non-localized line-solitons and localized solitons (dromions). Each soliton propagates with the group velocity and during a collision maintains its shape, the only change being a phase shift.

Interface dynamics in hele-shaw flows with centrifugal forces: preventing cusp singularities with rotation

A class of exact solutions of Hele-Shaw flows without surface tension in a rotating cell is reported. We show that the interplay between injection and rotation modifies the scenario of formation of finite-time cusp singularities. For a subclass of solutions, we show that, for any given initial condition, there exists a critical rotation rate above which cusp formation is suppressed. We also find an exact sufficient condition to avoid cusps simultaneously for all initial conditions within the above subclass.

Non-resonant interacting ion acoustic waves in a magnetized plasma

We perform an analytical and numerical investigation of the interaction among non-resonant ion acoustic waves in a magnetized plasma. Waves are supposed to be non-resonant, i.e. with different group velocities that are not close to each other. We use an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. We show that the amplitude slow modulation of Fourier modes cannot be described by the usual nonlinear Schrodinger equation but by a new model system of nonlinear evolution equations. This system is C-integrable, i.e. it can be linearized through an appropriate transformation of the dependent and independent variables. We demonstrate that a subclass of solutions gives rise to envelope solitons. Each envelope soliton propagates with its own group velocity. During a collision solitons maintain their shape, the only change being a phase shift. Numerical results are used to check the validity of the asymptotic perturbation method.

The Character-Automorphic Nehari Problem: Non-Uniqueness Criterion and some Extremal Solutions

A non-uniqueness criterion for the character-automorphic Nehari problem is given. Certain subclass of solutions, connected with "the entropy functional" of the problem, is described. The description yields a character-automorphic counterpart of the Adamyan-Arov-Krein theorem.