Nonlinear ordinary differential equations: A discussion on symmetries and singularities

Paliathanasis, Andronikos

doi:10.1142/s0219887816300099

Cited by 45 publications

(63 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [19], [20] it has been shown that in these case the Lie point symmetries are generated by the special projective algebra and the Noether point symmetries by the Homothetic algebra of the kinetic metric. Similar results have been found for some partial differential equations of special interest in curved spacetimes, as the wave and the heat equation (see [21], [22], [23] and references therein) where it has been shown that in these cases the Lie point symmetries involve the CKVs.…”

Section: Introductionsupporting

confidence: 80%

“…We conclude that the Bianchi III spacetime ds 2 (III) = e mλτ A 2 (τ ) −dτ 2 + dx 2 + e −mτ dy 2 + e −mλτ e −2x dz 2 (22) admits the proper CKV L 1 with conformal factor ψ (III) (L 1 ) = 2 m A,τ A + λ which reduces to a HV when A (τ ) is an exponential in which case the line element is ds 2 (III) = −e mκτ dτ 2 + e mκτ dx 2 + e m(κ−1)τ dy 2 + e m(κ−λ)τ e −2x dz 2 (23) or in equivalent form…”

Section: Case γ 2 (τ ) = E Mτmentioning

confidence: 68%

See 1 more Smart Citation

Constructing the CKVs of Bianchi III and V spacetimes

2019

Self Cite

View full text Add to dashboard Cite

We determine the conformal algebra of Bianchi III and Bianchi V spacetimes or, equivalently, we determine all Bianchi III and Bianchi V spacetimes which admit a proper conformal Killing vector. The algorithm that we use has been developed in Class. Quantum. Grav. 15, 2909 and concerns the computation of the CKVs of decomposable spacetimes. The main point of this method is that a decomposable space admits a CKV if the reduced space admits a gradient homothetic vector the latter being possible only if the reduced space is flat or a space of constant curvature. We apply this method in a stepwise manner starting from the two dimensional spacetime which admits an infinite number of CKVs and we construct step by step the Bianchi III and V spacetimes by assuming that CKVs survive as we increase the dimension of the space. We find that there is only one Bianchi III and one Bianchi V spacetime which admit at maximum one proper CKV. In each case we determine the conformal Killing vector and the corresponding conformal factor. As a first application in these two spacetimes we study the kinematics of the comoving observers and the dynamics of the corresponding cosmological fluid. As a second application we determine in these spacetimes generators of the Lie symmetries of the wave equation.

show abstract

Section: Introductionsupporting

confidence: 80%

Section: Case γ 2 (τ ) = E Mτmentioning

confidence: 68%

Constructing the CKVs of Bianchi III and V spacetimes

2019

Self Cite

View full text Add to dashboard Cite

show abstract

“…One can see that the above equation is invariant under time translations. For that if we select H = v and dH/dt = W (v), then the differential equation becomes the first order Abel equation of the second type: W W v + AW + Bv 2 + Cv + D = 0, where W v = dW/dv, and its solution in general cannot be written in a closed form expression, for more discussions see [38]. Thus, in the following we we shall consider several models starting from the simplest to the complex one and their physical consequences as well.…”

Section: Model 2: λ(Hḣḧ)mentioning

confidence: 99%

“…Since α is unrestricted, thus we consider two separate cases, namely, α < 0 and α > 0. Now, solving the equation (38) for α < 0, one finds that the Hubble factor evolves as > 0 as H 0 > 0. The scale factor is can be solved as…”

Section: The Model λ(ḣ) = α + µḣmentioning

confidence: 99%

Exact solutions, finite time singularities and non-singular universe models from a variety of Λ(t) cosmologies

Pan

2018

Mod. Phys. Lett. A

View full text Add to dashboard Cite

Cosmological models with time dependent Λ (read as Λ(t)) have been investigated widely in the literature. Models that solve background dynamics analytically, are of special interest. Additionally, the allowance of past or future singularities at finite cosmic time in a specific model signals for a generic test on its viabilities with the current observations. Following these, in this work we consider a variety of Λ(t) models focusing on their evolutions and singular behaviour. We found that a series of models in this class can be exactly solved when the background universe is described by a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) line element. The solutions in terms of the scale factor of the FLRW universe offer different universe models, such as power law expansion, oscillating, and the singularity free universe as well. However, we also noticed that a large number of models in this series permits past or future cosmological singularities at finite cosmic time. At last we close the work with a note that the avoidance of future singularities is possible for certain models under some specific restrictions.PACS numbers: 98.80.-k, 95.36.+x

show abstract

“…Two of the most famous are: (a) the existence of invariant transformations, i.e. symmetries and (b) the singularity analysis; for a recent discussion and comparison of these two methods see [10]. Both of these have been applied widely in gravitational theories [11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Integrability from point symmetries in a family of cosmological Horndeski Lagrangians

2017

Self Cite

View full text Add to dashboard Cite

For a family of Horndeski theories, formulated in terms of a generalized Galileon model, we study the integrability of the field equations in a Friedmann-Lemaître-Robertson-Walker space-time. We are interested in point transformations which leave invariant the field equations. Noether's theorem is applied to determine the conservation laws for a family of models that belong to the same general class. The cosmological scenarios with or without an extra perfect fluid with constant equation of state parameter are the two important cases of our study. The de Sitter universe and ideal gas solutions are derived by using the invariant functions of the symmetry generators as a demonstration of our result. Furthermore, we discuss the connection of the different models under conformal transformations while we show that when the Horndeski theory reduces to a canonical field the same holds for the conformal equivalent theory. Finally, we discuss how singular solutions provides nonsingular universes in a different frame and vice versa.

show abstract

Nonlinear ordinary differential equations: A discussion on symmetries and singularities

Abstract: Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different methods are given.

Cited by 45 publications

References 49 publications

Constructing the CKVs of Bianchi III and V spacetimes

Constructing the CKVs of Bianchi III and V spacetimes

Exact solutions, finite time singularities and non-singular universe models from a variety of Λ(t) cosmologies

Integrability from point symmetries in a family of cosmological Horndeski Lagrangians

Contact Info

Product

Resources

About