From Feshbach-resonance managed Bose–Einstein condensates to anisotropic universes: Applications of the Ermakov–Pinney equation with time-dependent nonlinearity

Herring, G.; Kevrekidis, P. G.; Williams, Floyd L.; Christodoulakis, T.; Frantzeskakis, D. J.

doi:10.1016/j.physleta.2007.01.078

Cited by 3 publications

(1 citation statement)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Ermakov-Pinney equation has been the focus of many research studies of nonlinear dynamical systems, for a survey see [257]. The equation also arises in variety of physical situation for instance in scalar field cosmology [258], Feshbach-resonance of BECs [259], elasticity [260,261], quantum mechanics [262].…”

Section: A Simple Casementioning

confidence: 99%

Renormalization Group as a Probe for Dynamical Systems

Sarkar

Bhattacharjee

2011

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

Section: A Simple Casementioning

confidence: 99%

Renormalization Group as a Probe for Dynamical Systems

Sarkar

Bhattacharjee

2011

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

Cosmological aspects of the Eisenhart–Duval lift

et al. 2018

View full text Add to dashboard Cite

A cosmological extension of the Eisenhart-Duval metric is constructed by incorporating a cosmic scale factor and the energy-momentum tensor into the scheme. The dynamics of the spacetime is governed the Ermakov-Milne-Pinney equation. Killing isometries include spatial translations and rotations, Newton-Hooke boosts and translation in the null direction. Geodesic motion in Ermakov-Milne-Pinney cosmoi is analyzed. The derivation of the Ermakov-Lewis invariant, the Friedmann equations and the Dmitriev-Zel'dovich equations within the Eisenhart-Duval framework is presented.

show abstract

The Ermakov equation: A commentary

Leach

Andriopoulos

2008

APPL ANAL DISCRETE M

115

127

View full text Add to dashboard Cite

We present a short history of the Ermakov Equation with an emphasis on its discovery by the West and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008), 123-145) for an English translation of Ermakov's original paper. THE LEWIS INVARIANT AND PINNEY'S SOLUTIONIn 1950 the late Edmund Pinney [2] presented in a very succinct paper [85] the solution of the equationin which the overdot represents differentiation with respect to the independent variable, t, which in many applications is the time (See also [7]). The solution which he gave is(2) x(t) = Au 2 + 2Buv + Cv 2 1/2 , where u(t) and v(t) are any two linearly independent solutions of the equation0 2000 Mathematics Subject Classification. 34A05, 01A75, 70F05, 22E70. The Ermakov Equation: history and impact 147 and the constants, A, B and C, are related according to B 2 − AC = 1/W 2 with W being the constant Wronskian of the two linearly independent solutions.In 1966 the late Ralph Lewis, while he was on sabbatical at the University of Heidelberg, commenced the calculation of an invariant for the Hamiltonian corresponding to (3), videlicetusing Kruskal's asymptotic method [54]. An adiabatic invariant for (4) had been proposed by Lorentz at the Solvay Congress of 1911 [98], but this was not satisfactory for the application of interest to Lewis which was the motion of a charged particle in an electromagnetic field expected to be rapidly varying in the context of plasma confinement. Kruskal's method involves an asymptotic expansion in terms of a parameter, ε, and for the first term in the expansion Lewis obtainedwhere ρ(t) is a solution of (1) and so is given by (2). To the surprise of Lewis the second and third terms in the asymptotic expansion were zero. He then essayed 1 a direct calculation of the first derivative of I 0 and found it to be zero! His results are found in [65][66][67].Although this result could scarcely be considered to be of use in classical problems, the reduction of the solution of the corresponding time-dependent Schrödinger equation [68] to the solution of (3) could be regarded as a distinct advantage since any numerical computation was then deferred until almost the last line. A clear example of this is found in the calculation of Berry's Phase for the time-dependent linear oscillator [55]. The utility of the result was enhanced when it was found that (1) occurred as an auxiliary equation in the calculation of invariants for nonquadratic Hamiltonian systems [16,17,37,38,26,[69][70][71][72][73][74]. ERMAKOV AND HIS INVARIANTIn Ermakov's paper the roles of (1), (3) and (5) are interchanged by comparison with the work of Lewis 2 . Ermakov introduces (1) as an equation auxiliary to (3), multiplies by an integr...

show abstract

From Feshbach-resonance managed Bose–Einstein condensates to anisotropic universes: Applications of the Ermakov–Pinney equation with time-dependent nonlinearity

Cited by 3 publications

References 48 publications

Renormalization Group as a Probe for Dynamical Systems

Renormalization Group as a Probe for Dynamical Systems

Cosmological aspects of the Eisenhart–Duval lift

The Ermakov equation: A commentary

Contact Info

Product

Resources

About