Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{\zeta \zeta}+g(v)v_{\zeta}+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h_0(v) is a linear function, h_0(v)=\lambda^2v, 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(v)=\Omega_0^2(v-v^2) and show that it leads to an integrable hyperelliptic caseComment: 15 pages, 5 figures, 1 appendix, 21 references, published versio

Ermakov-Lewis Invariants and Reid Systems

Reid's m'th-order generalized Ermakov systems of nonlinear coupling constant alpha are equivalent to an integrable Emden-Fowler equation. The standard Ermakov-Lewis invariant is discussed from this perspective, and a closed formula for the invariant is obtained for the higher-order Reid systems (m\geq 3). We also discuss the parametric solutions of these systems of equations through the integration of the Emden-Fowler equation and present an example of a dynamical system for which the invariant is equivalent to the total energyComment: 8 pages, published versio

One-Parameter Darboux-Deformed Fibonacci Numbers

One-parameter Darboux deformations are effected for the simple ODE satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (non integer) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov-Lewis invariants for these sequences are also discussed

One-parameter Darboux-deformed Fibonacci numbers

One-parameter Darboux deformations are effected for the simple ODE satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (non integer) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov-Lewis invariants for these sequences are also discussed.Comment: 7 pages, 4 figure

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects

Squeezing equivalence of quantum harmonic oscillators under different frequency jumps

In their studies on the squeezing produced by a sequence of sudden frequency changes of a quantum harmonic oscillator, Janszky and Adam [Phys. Rev. A {\bf 46}, 6091 (1992)] found the following equivalence: a harmonic oscillator, under a sequence of two sudden frequency jumps, from $\omega_0$ to $\omega_1$ and back to $\omega_0$ (after a time interval $\tau$), exhibits, for $\tau=k\pi/\omega_1$ ($k\in\mathbb{N}$), exactly the same squeezing parameter as the harmonic oscillator whose frequency would remain constant [specifically, $r(t>\tau)=0$]. In the present paper, we show an extended version of this equivalence, demonstrating how to set up different sequences of two sudden frequency jumps, so that, despite having different intermediate frequencies during a time interval $\tau$, they result in a same value $r(t>\tau)\neq0$ (and, consequently, in the same physical quantities that depend on it) after the jumps cease. Applied to a particular situation, our formulas recover the equivalence obtained by Janszky and Adam.Comment: 10 pages, 6 figure