Hybrid Ermakov-Painlevé IV Systems

Rogers, C.

doi:10.1080/14029251.2014.975531

Cited by 12 publications

(7 citation statements)

References 58 publications

(57 reference statements)

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“…Our findings reveals the importance of future studies devoted to construct analytical/approximated solutions of the Gambier equations in order to better understand the behaviour of Gambier inspired cosmological models. Even though, the role of the Ermakov-Pinney equation in FLRW cosmology is well known in the literature, here we have explored the role of various generalizations of Pinney equation, as formulated by [26][27][28], in 2+1-dimensional FLRW cosmology. This raises an important issue regarding the existence of a systematic relation between integrable class of 1+1-ODEs and FLRW cosmology and clearly, demands further investigation.…”

Section: Discussionmentioning

confidence: 99%

“…He showed that the Ermakov invariants admitted by the hybrid system were key to its systematic reduction in terms of a single component Ermakov-Painlevé II equation which, in turn, may be linked to the integrable Painlevé II equation. Rogers [27] further extended the hybrid family to derive Ermakov-Painlevé IV equation and investigated the connection between the FLRW cosmology of [24] and the generalized Pinney equations derived by Rogers, Schief and Winternitz [28]. We may exploit the underlying Ermakov type invariants of these equations to obtain solutions of the reduced FLRW cosmological equation with scalar field and perfect fluid matter source.…”

Section: Introductionmentioning

confidence: 99%

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Emergence of the Gambier equation in cosmology

2023

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In this paper, we show how the Gambier equation arises in connection to Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology and a Dark Matter equation of state. Moreover, we provide a correspondence between the Friedmann equations and the Gambier equations that possess the Painlevé property in (2 + 1) dimensions. We also consider special cases of the Gambier G27 equation such as the generalized Pinney equation. For an extended FLRW model with dynamic scalar field as matter model, the Einstein equations correspond to the Milne–Pinney equation which in turn can be mapped to the parametric Gambier equation of second order.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Emergence of the Gambier equation in cosmology

2023

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“…In reference [4] the Painlevé property test was established in the context of equation (1) with arbitrary ǫ i , i = 0, 1. The idea of of generalizing two different Painlevé equations has been explored in the literature in different settings and good examples of such work are [15], [25].…”

Section: Review Of P Iii−v Equations and Their Symmetriesmentioning

confidence: 99%

Solutions of mixed Painlevé P_III—Vmodel

Alves

Aratyn

Gomes

et al. 2019

J. Phys. A: Math. Theor.

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We review the construction of the mixed Painlevé P III−V system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its parameters reduces to either Painlevé P III or P V . The aim of this paper is to describe solutions of P III−V model. In particular, we determine and classify rational, power series and transcendental solutions of P III−V . A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P III−V equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order n when the parameter of P III−V equation is quantized by integer n ∈ Z.Painlevé P II equation. Painlevé P V equations appeared in the context of impenetrable Bose gas model [13]. More recently the scattering on two Aharonov-Bohm vortices was exactly solved in terms of solutions of the Painlevé III equation [7]. Ablowitz Ramani and Segur (ARS) showed a connection between the mKdV integrable model and the Painlevé P II equation when self-similarity limit was implemented. They suggested that such connection could be generalized to other integrable models [1] meaning that the self-similarity limit would lead for integrable models to equations with the Painlevè property. More recently, it was found [3] that the 2n boson integrable model obtained as particular reductions (Drinfeld-Sokolov) of KP integrable models connected to Toda lattice hierarchy gives rise to higher Painlevé equations invariant under extended affine Weyl groups. In particular, investigation of various Dirac reduction schemes applied to the 4-boson (n = 2) integrable model with Weyl symmetry structure A (1) 4 led to emergence of a new mixed P III−V model [4]. The second order P III−V equation for a canonical variable q is given by:

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“…Thus, the only known hybrid solitonic-Ermakov system seems to be that obtained in [54,55] where a 2+1-dimensional Ernst-type system of general relativity as derived in [56], suitably constrained, leads to a novel composition of the integrable 2+1-dimensional sinh-Gordon equation of [30,31] and of a generalised Ermakov-Ray-Reid system. The work of [2,[36][37][38] on Ermakov-Painlevé II systems has recently been augmented by the introduction in [47] of prototype Ermakov-Painlevé IV systems via a symmetry reduction of a coupled derivative resonant NLS triad. Dirichlet type two-point boundary value problems for a single hybrid Ermakov-Painlevé IV equation have been investigated with regard to existence and uniqueness properties in [3].…”

Section: Introductionmentioning

confidence: 99%