The damped Pinney equation and its applications to dissipative quantum mechanics

a b s t r a c tWe introduce a special type of dissipative Ermakov-Pinney equations of the form v ff þ gðvÞv f þ hðvÞ ¼ 0, where hðvÞ ¼ h 0 ðvÞ þ cv À3 and the nonlinear dissipation gðvÞ is based on the corresponding Chiellini integrable Abel equation. When h 0 ðvÞ is a linear function, h 0 ðvÞ ¼ k 2 v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipative function is actually a dissipation-gain function because it can be negative on some intervals. We also examine the nonlinear case h 0 ðvÞ ¼ X 2 0 ðv À v 2 Þ and show that it leads to an integrable hyperelliptic case.

show abstract

“…Furthermore, using the substitution v ζ ≡ 1/y(v) between (10) and (12) in (14) and integrating, one has…”

Section: B Chiellini Dissipative Sep (Cd-sep) Equationsmentioning

confidence: 99%

Integrable equations with Ermakov–Pinney nonlinearities and Chiellini damping

Mancas

Rosu

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

show abstract

“…In the conservative (ν = 0) case the (nonlinear) Pinney equation is well known to be exactly solvable in terms of the solution of the (linear) time-dependent harmonic oscillator equation [15]. Approximate solutions exist [5] for weak damping and slowly varying non-vanishing frequencies ω(t).…”

Section: Expansion Around Classical Trajectoriesmentioning

confidence: 99%

“…Moreover it can be checked that the simple approximate expressions from Ref. [5] developed for a non-zero harmonic confinement did not apply in the free case.…”

Section: The Free Particle Casementioning

confidence: 99%

Time-Dependent Gaussian Solution for the Kostin Equation Around Classical Trajectories

et al. 2012

View full text Add to dashboard Cite

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.

show abstract

“…It is known that as ζ = 1, the Pinney equation has the following particular solution (see e.g. [21])…”

Section: Pinney Cosmologymentioning

confidence: 99%

Dark Energy in Some Integrable and Nonintegrable FRW Cosmological Models

Esmakhanova

Myrzakulov

Nugmanova

et al. 2011

Int. J. Mod. Phys. D

View full text Add to dashboard Cite

One of the greatest challenges in cosmology today is to determine the nature of dark energy, the sourse of the observed present acceleration of the Universe. Besides the vacuum energy, various dark energy models have been suggested. The Friedmann -Robertson -Walker (FRW) spacetime plays an important role in modern cosmology. In particular, the most popular models of dark energy work in the FRW spacetime. In this work, a new class of integrable FRW cosmological models is presented. These models induced by the well-known Painlevé equations. Some nonintegrable FRW models are also considered. These last models are constructed with the help of Pinney, Schrödinger and hypergeometric equations. Scalar field description and two-dimensional generalizations of some cosmological models are presented. Finally some integrable and nonintegrable F (R) and F (G) gravity models are constructed.

show abstract

The damped Pinney equation and its applications to dissipative quantum mechanics

Cited by 20 publications

References 46 publications

Integrable equations with Ermakov–Pinney nonlinearities and Chiellini damping

Integrable equations with Ermakov–Pinney nonlinearities and Chiellini damping

Time-Dependent Gaussian Solution for the Kostin Equation Around Classical Trajectories

Dark Energy in Some Integrable and Nonintegrable FRW Cosmological Models

Contact Info

Product

Resources

About