Complex Riccati equations as a link between different approaches for the description of dissipative and irreversible systems

115

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finitedimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known Lie-Hamilton systems on R 2 with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach. MSC: 34A26 (primary) 70G45, 70H99 (secondary)Hence, system (1.4) with a R 1 (t) = 0 is a LH system as it is related to a t-dependent vector field taking values in a Vessiot-Guldberg Lie algebra V of Hamiltonian vector fields relative to Λ. Since V ≃ R ⋉ R 2 ≃ X 1 ⋉ X 2 , X 3 , X 1 ∧ X 2 = 0 and ad X 1 : X i ∈ X 2 , X 3 → [X 1 , X i ] ∈ X 2 , X 3 is not diagonalizable over R, we see in view of table 1 that the Lie algebra V belongs to class P 1 and V ≃ iso(2). Meanwhile, the LH algebra spanned by h 1 , h 2 , h 3 , h 0 is isomorphic to the centrally extended Euclidean algebra iso(2) (see also [13] for further details).On the other hand, we recall that superposition rules for LH systems can be obtained in an algebraic way by applying a Poisson coalgebra approach [19]. In contrast, other methods to derive superposition rules require to integrate a Vessiot-Guldberg Lie algebra, e.g., the group theoretical method [2], or to solve a family of PDEs [9]. Winternitz and coworkers have also derived superposition rules for Lie systems in particular forms [2,5]. The application of this latter result for general Lie systems requires to map them into the canonical form for which the superposition rule was obtained. The coalgebra procedure makes these transformations unnecessary in many cases.The structure of the paper is as follows. In section 2 we summarize the local classification of Vessiot-Guldberg Lie algebras as of Hamiltonian vector fields on the plane performed in [13], where the corresponding symplectic structures were derived by solving a system of PDEs. As a first new achievement, we show in section 3 that such symplectic structures can be determined through algebraic and geometric methods. Although our techniques are heavily based upon the Lie algebra structure of Vessiot-Guldberg Lie algebras, they also depend on their geometric properties as Lie algebras of vector fields. In particular, a new method to construct symplectic structures for LH systems on the plane related to non-simple Vessiot-Guldberg Lie algebras is described.Next we remark that a Vessiot-Guldberg Lie algebra on the plane isomorphic to sl(2) can be diffeomorphic to either P 2 , I 3 ...

Section: Cayley-klein Riccati Equationsmentioning

confidence: 99%

Lie–Hamilton systems on the plane: applications and superposition rules

Blasco¹,

Herranz²,

Lucas³

et al. 2015

115

“…now with a + (t ) as defined in (21b) (or (26b)). Evaluating e za + with the help of z and |z| 2 , as given in (33) and (34), leads again in the position-space representation to z (x, t ) as given in (37).…”

Section: Generalized Creation and Annihilation Operators And Correspo...mentioning

confidence: 99%

Generalized coherent states for time-dependent and nonlinear Hamiltonian operators via complex Riccati equations

Castaños

Schuch

Rosas-Ortiz

2013

Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex variable that fulfills a nonlinear Riccati equation which determines the time-evolution of the wave packet width. The solutions of this Riccati equation depend on the physical system under consideration and on the (complex) initial conditions and have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics. The definition of the generalized creation and annihilation operator is also in agreement with a factorization of the operator corresponding to the Ermakov invariant that exists in all cases considered.

“…where b i (t) are arbitrary t-dependent real coefficients. We recall that (5.1) is related to certain planar Riccati equations [49,50] and that several mathematical and physical applications can be found in [66,67,68,69]. By writing z = u + iv, we find that (5.1) gives rise to a system of the type (2.1), namely…”

Section: Deformed Complex Riccati Equationmentioning

confidence: 99%

Poisson–Hopf algebra deformations of Lie–Hamilton systems

Ballesteros¹,

Campoamor-Stursberg²,

Fernández‐Saiz³

et al. 2018

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.