Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie-Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie-Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie-Hamilton system associated with a Poisson-Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson-Hopf algebra analogue of the non-standard quantum deformation of sl(2) and its applications to deform well-known Lie-Hamilton systems describing oscillator systems, Milne-Pinney equations, and several types of Riccati equations. In particular, we obtain a new positiondependent mass oscillator system with a time-dependent frequency.
Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems [4], a novel unified approach to nonequivalent deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to sl(2) is proposed. This, in particular, allows us to define a notion of Poisson-Hopf systems in dependence of a parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of sl(2) Lie-Hamilton systems. Our results cover deformations of the Ermakov system, Milne-Pinney, Kummer-Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with t-dependent coefficients). Furthermore t-independent constants of motion are given as well. Our methods can be employed to generate other Lie-Hamilton systems and their deformations for other Vessiot-Guldberg Lie algebras and their deformations. 1
In a recent paper (Nakamura and Martinez, 2019), the classical epidemic compartmental susceptible-infectious-susceptible (SIS) model has been upgraded to a form which permits fluctuations in terms of the mean and the variance of infected individuals. This novel model happens to admit a Hamiltonian realization involving a constant 𝜌 0 that is related to the reproduction number R 0 . In this work, we generalize 𝜌 0 as a t-dependent function to arrive at a non-autonomous system of differential equations that we call SISt model. We make use of the theory of Lie systems to provide a (nonlinear) superposition rule for the general solution of SISt model. To conclude, we present a quantum version of the same model, giving rise to a parametric family of SISt systems. We provide a superposition rule for the general solution of the quantum extension as well.
<abstract><p>Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that extend a recently proposed SIS epidemic model with a variable infection rate are considered. It is shown that, independently on the particular interpretation of the time-dependent coefficients, these systems generally admit an exact solution, up to the case of the maximal extension within the classification of Lie-Hamilton systems, for which a superposition rule is constructed. The method provides the algebraic frame to which any SIS epidemic model that preserves the above-mentioned properties is subjected. In particular, we obtain exact solutions for generalized SIS Hamiltonian models based on the book and oscillator algebras, denoted by $ \mathfrak{b}_2 $ and $ \mathfrak{h}_4 $, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry $ \mathfrak{h}_6 $, according to the embedding chain $ \mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6 $, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.</p></abstract>
The formalism for Poisson–Hopf (PH) deformations of Lie–Hamilton (LH) systems, recently proposed in Ballesteros Á et al (2018 J. Phys. A: Math. Theor. 51 065202), is refined in one of its crucial points concerning applications, namely the obtention of effective and computationally feasible PH deformed superposition rules for prolonged PH deformations of LH systems. The two new notions here proposed are a generalization of the standard superposition rules and the concept of diagonal prolongations for Lie systems, which are consistently recovered under the non-deformed limit. Using a technique from superintegrability theory, we obtain a maximal number of functionally independent constants of the motion for a generic prolonged PH deformation of a LH system, from which a simplified deformed superposition rule can be derived. As an application, explicit deformed superposition rules for prolonged PH deformations of LH systems based on the oscillator Lie algebra h 4 are computed. Moreover, by making use that the main structural properties of the book subalgebra b 2 of h 4 are preserved under the PH deformation, we consider prolonged PH deformations based on b 2 as restrictions of those for h 4 -LH systems, thus allowing the study of prolonged PH deformations of the complex Bernoulli equations, for which both the constants of the motion and the deformed superposition rules are explicitly presented.
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