P. G. L. Leach
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
P. G. L. Leach has a total of 8 co-authors. Here are the top co-authors by number of shared publications:
- Andronikos Paliathanasis
Durban University of Technology
8 shared publications - Christodoulos Sophocleous
University of Cyprus
2 shared publications - John D. Barrow
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
2 shared publications - Sameerah Jamal
University of the Witwatersrand
2 shared publications - Tim Taves
Centro de Estudios Científicos (CECs) Arturo Prat 514 Valdivia Chile
2 shared publications - Olena O. Vaneeva
Institute of Mathematics NAS of Ukraine 3 Tereshchenkivska Str. Kyiv-4 01601 Ukraine
1 shared publication
Latest Publications
- 16 March 2018
- Published by National Inquiry Services Center (NISC) in Quaestiones Mathematicae
- Vol. 42 (1) , 125-133
Nonlocal representation of the sl(2, R) algebra for the Chazy equation
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.
- 16 March 2018
- Published by National Inquiry Services Center (NISC) in Quaestiones Mathematicae
- Vol. 42 (1) , 125-133
Nonlocal representation of the sl(2, R) algebra for the Chazy equation
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.
- Open AccessNonlocal Representation of the $sl(2,R)$ Algebra for the Chazy equation
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 Painlev\'e 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 Painlev\'e Series and one given by a Right Painlev\'e Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.
Open AccessNonlocal Representation of the $sl(2,R)$ Algebra for the Chazy equationA 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 Painlev\'e 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 Painlev\'e Series and one given by a Right Painlev\'e Series where the leading-order behaviors and the resonances are explicitly those of the Chazy Equation.
- Open AccessIntegrability of the Einstein-nonlinear $SU(2)$ $σ$-model in a nontrivial topological sectorThe integrability of the\ $\Lambda-$Einstein-nonlinear $SU(2)$ $\sigma$-model with nonvanishing cosmological charge is studied. We apply the method of singularity analysis of differential equations and we show that the equations for the gravitational field are integrable. The first few terms of the solution are presented.Open AccessIntegrability of the Einstein-nonlinear $SU(2)$ $σ$-model in a nontrivial topological sectorThe integrability of the\ $\Lambda-$Einstein-nonlinear $SU(2)$ $\sigma$-model with nonvanishing cosmological charge is studied. We apply the method of singularity analysis of differential equations and we show that the equations for the gravitational field are integrable. The first few terms of the solution are presented.
- Open Access
- 27 December 2017
- Published by Springer Nature in The European Physical Journal C
- Vol. 77 (12) , 909
Integrability of the Einstein-nonlinear SU(2) $\sigma $ σ -model in a nontrivial topological sector- 27 December 2017
- Published by Springer Nature in The European Physical Journal C
- Vol. 77 (12) , 909
Open AccessIntegrability of the Einstein-nonlinear SU(2) $\sigma $ σ -model in a nontrivial topological sector - 28 July 2016
- Published by American Physical Society (APS) in Physical Review D
- Vol. 94 (2) , 023525
Cosmological solutions ofgravityIn the cosmological scenario in gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear -gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an theory.- 28 July 2016
- Published by American Physical Society (APS) in Physical Review D
- Vol. 94 (2) , 023525
Cosmological solutions ofgravityIn the cosmological scenario in gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear -gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an theory.- Open AccessCosmological Solutions of $f(T)$ GravityIn the cosmological scenario in $f\left( T\right) $ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear $f\left( T\right) $-gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an $f(T)$ theory.Open AccessCosmological Solutions of $f(T)$ GravityIn the cosmological scenario in $f\left( T\right) $ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear $f\left( T\right) $-gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an $f(T)$ theory.
- Open AccessAnalytical solutions in $R+qR^{n}$ cosmology from singularity analysisThe integrability of higher-order theories of gravity is of importance in the determining the properties of these models and so their viability as models of reality. An important tool in the establishment of integrability is the singularity analysis. We apply this analysis to the case of fourth-order theory of gravity $f(R) = R + qR^{n}$ to establish those values of the free parameters $q$ and $n$ for which integrability in this sense exists. As a prelininary we examine the well-known case of $n = 4/3$.Open AccessAnalytical solutions in $R+qR^{n}$ cosmology from singularity analysisThe integrability of higher-order theories of gravity is of importance in the determining the properties of these models and so their viability as models of reality. An important tool in the establishment of integrability is the singularity analysis. We apply this analysis to the case of fourth-order theory of gravity $f(R) = R + qR^{n}$ to establish those values of the free parameters $q$ and $n$ for which integrability in this sense exists. As a prelininary we examine the well-known case of $n = 4/3$.
- Open Access
On the Hojman conservation quantities in Cosmology
We discuss the application of the Hojman's Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman's method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether's Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like $f(R)$ gravity, the application of Hojman's method provide us with the same results with that of Noether's theorem. Moreover we study the special Ansatz. $\phi\left( t\right) =\phi\left( a\left( t\right) \right) $, which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of Noetherian conservation laws for the free particle. On the other hand, for $f(T)$ teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of $f(T)$ functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation.Open AccessOn the Hojman conservation quantities in Cosmology
We discuss the application of the Hojman's Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman's method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether's Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like $f(R)$ gravity, the application of Hojman's method provide us with the same results with that of Noether's theorem. Moreover we study the special Ansatz. $\phi\left( t\right) =\phi\left( a\left( t\right) \right) $, which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of Noetherian conservation laws for the free particle. On the other hand, for $f(T)$ teleparallel gravity, it is not the existence of Hojman's conservation laws which provide us with the special function form of $f(T)$ functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation. - 22 November 2014
- Published by Springer Nature in Journal of Engineering Mathematics
- Vol. 91 (1) , 165-176
Lie symmetries of generalized Burgers equations: application to boundary-value problems
- 22 November 2014
- Published by Springer Nature in Journal of Engineering Mathematics
- Vol. 91 (1) , 165-176
Lie symmetries of generalized Burgers equations: application to boundary-value problems
- Open Access
Lie symmetries of generalized Burgers equations: application to boundary-value problems
There exist several approaches exploiting Lie symmetries in the reduction of boundary-value problems for partial differential equations modelling real-world phenomena to those problems for ordinary differential equations. Using an example of generalized Burgers equations appearing in nonlinear acoustics we show that that the "direct" procedure of solving boundary-value problems using Lie symmetries firstly described by Bluman is more general and straightforward than the method suggested by Moran and Gaggioli in [J. Eng. Math. 3 (1969), 151-162]. After the group classification of a class of generalized Burgers equations with time-dependent viscosity is performed we solve an associated boundary-value problem using the symmetries obtained.
Open AccessLie symmetries of generalized Burgers equations: application to boundary-value problems
There exist several approaches exploiting Lie symmetries in the reduction of boundary-value problems for partial differential equations modelling real-world phenomena to those problems for ordinary differential equations. Using an example of generalized Burgers equations appearing in nonlinear acoustics we show that that the "direct" procedure of solving boundary-value problems using Lie symmetries firstly described by Bluman is more general and straightforward than the method suggested by Moran and Gaggioli in [J. Eng. Math. 3 (1969), 151-162]. After the group classification of a class of generalized Burgers equations with time-dependent viscosity is performed we solve an associated boundary-value problem using the symmetries obtained.