Sameerah Jamal

Dynamical analysis of an integrable cubic Galileon cosmological model

Recently a cubic Galileon cosmological model was derived by the assumption that the field equations are invariant under the action of point transformations. The cubic Galileon model admits a second conservation law which means that the field equations form an integrable system. The analysis of the critical points for this integrable model is the main subject of this work. To perform the analysis, we work on dimensionless variables different from that of the Hubble normalization. New critical points are derived while the gravitational effects which follow from the cubic term are studied.

Noether symmetries and stability of ideal gas solutions in Galileon cosmology

A class of generalized Galileon cosmological models, which can be described by a point-like Lagrangian, is considered in order to utilize Noethers Theorem to determine conservation laws for the field equations. In the Friedmann-Lema\^{\i}tre-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Due to the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.

Group invariant transformations for the Klein–Gordon equation in three dimensional flat spaces

Abstract We perform the complete symmetry classification of the Klein–Gordon equation in maximal symmetric spacetimes. The central idea is to find all possible potential functions V ( t , x , y ) that admit Lie and Noether symmetries. This is done by using the relation between the symmetry vectors of the differential equations and the elements of the conformal algebra of the underlying geometry. For some of the potentials, we use the admitted Lie algebras to determine corresponding invariant solutions to the Klein–Gordon equation. An integral part of this analysis is the problem of the classification of Lie and Noether point symmetries of the wave equation.

Approximate Noether symmetries and collineations for regular perturbative Lagrangians

Abstract Regular perturbative Lagrangians that admit approximate Noether symmetries and approximate conservation laws are studied. Specifically, we investigate the connection between approximate Noether symmetries and collineations of the underlying manifold. In particular we determine the generic Noether symmetry conditions for the approximate point symmetries and we find that for a class of perturbed Lagrangians, Noether symmetries are related to the elements of the Homothetic algebra of the metric which is defined by the unperturbed Lagrangian. Moreover, we discuss how exact symmetries become approximate symmetries. Finally, some applications are presented.

Noether symmetries of vacuum classes of pp-waves and the wave equation

The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noethers theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.

Potentials and point symmetries of Klein–Gordon equations in space-time homogenous Gödel-type metrics

In this paper, we study the geometric properties of generators for the Klein–Gordon equation on classes of space-time homogeneous Godel-type metrics. Our analysis complements the study involving the “Symmetries of geodesic motion in Godel-type spacetimes” by U. Camci (J. Cosmol. Astropart. Phys., doi:10.1088/1475-7516/2014/07/002). These symmetries or Killing vectors (KVs) are used to construct potential functions admitted by the Klein–Gordon equation. The criteria for the potential function originates from three primary sources, viz. through generators that are identically the Killing algebra, or with the KV fields that are recast into linear combinations and third, real subalgebras within the Killing algebra. This leads to a classification of the (1 + 3) Klein–Gordon equation according to the catalogue of infinitesimal Lie and Noether point symmetries admitted. A comprehensive list of group invariant functions is provided and their application to analytic solutions is discussed.

Solutions of quasi-geostrophic turbulence in multi-layered configurations

Abstract We consider quasi-geostrophic (QG) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differential systems of evolution equations. A two-layer QG model, in a simplified version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane QG point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography QG model of special interest, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven doublegyre ocean flow for a range of physically relevant features, such as lateral friction and the analogue parameters of the f-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.

Nonlocal representation of the sl(2, R) algebra for the Chazy equation

Abstract A demonstration of how the point symmetries of the Chazy equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy equation under a generalized transformation, and find the point symmetries of the Chazy equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy equation which is given by a right Painlevé series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a left Painlevé series and one given by a right Painlevé series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation.

Approximate conditions admitted by classes of the Lagrangian L=12(−u′2+u2)+ϵiGi(u,u′,u″)

We investigate a class of Lagrangians that admit a type of perturbed harmonic oscillator which occupies a special place in the literature surrounding perturbation theory. We establish explicit and generalized geometric conditions for the symmetry determining equations. The explicit scheme provided can be followed and specialized for any concrete perturbed differential equation possessing the Lagrangian. A systematic solution of the conditions generate nontrivial approximate symmetries and transformations. Detailed cases are discussed to illustrate the relevance of the conditions, namely (a) G1 as a quadratic polynomial, (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle and (c) an orbital equation from an embedded Reissner-Nordström black hole.We investigate a class of Lagrangians that admit a type of perturbed harmonic oscillator which occupies a special place in the literature surrounding perturbation theory. We establish explicit and generalized geometric conditions for the symmetry determining equations. The explicit scheme provided can be followed and specialized for any concrete perturbed differential equation possessing the Lagrangian. A systematic solution of the conditions generate nontrivial approximate symmetries and transformations. Detailed cases are discussed to illustrate the relevance of the conditions, namely (a) G1 as a quadratic polynomial, (b) the Klein–Gordon equation of a particle in the context of Generalized Uncertainty Principle and (c) an orbital equation from an embedded Reissner–Nordstrom black hole.