C. Sardón

From constants of motion to superposition rules for Lie–Hamilton systems

A Lie system is a non-autonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie–Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie–Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods for deriving in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer–Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky–Winternitz systems with time-dependent frequency.

LIE–HAMILTON SYSTEMS: THEORY AND APPLICATIONS

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie–Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearizability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.

Dirac–Lie systems and Schwarzian equations

Abstract A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to Diracs description of constrained systems, we introduce and analyze a particular class of Lie systems on Dirac manifolds, called Dirac–Lie systems, which are associated with ‘Dirac–Lie Hamiltonians’. Our results enable us to investigate constants of the motion, superposition rules, and other general properties of such systems in a more effective way. Several concepts of the theory of Lie systems are adapted to this ‘Dirac setting’ and new applications of Dirac geometry in differential equations are presented. As an application, we analyze solutions of several types of Schwarzian equations, but our methods can be applied also to other classes of differential equations important for Physics.

Lie–Hamilton systems on the plane: Properties, classification and applications

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in (A. Gonzalez-Lopez, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)) and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

On Lie systems and Kummer-Schwarz equations

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer-Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,R). This same result can be extended to Riccati, Milne-Pinney, and to the here defined generalised Kummer-Schwarz equations, which include several types of Kummer-Schwarz equations as particular cases. We demonstrate that all the above-mentioned equations related to the same Lie system on SL(2,R) can be integrated simultaneously, which retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.

Lie symmetries for Lie systems

A Lie system is a nonautonomous system of first-order differential equations admitting a superposition rule, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. Using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra, we associate every Lie system with a Lie algebra of Lie point symmetries induced by the Vessiot-Guldberg Lie algebra. This enables us to derive Lie symmetries of relevant physical systems described by first- and higher-order systems of differential equations by means of Lie systems in an easier way than by standard methods. A generalization of our results to partial differential equations is introduced. Among other applications, Lie symmetries for several new and known generalizations of the real Riccati equation are studied.

Non-isospectral 1+1 hierarchies arising from a Camassa--Holm hierarchy in 2+1 dimensions

The non-isospectral problem (Lax pair) associated with a hierarchy in 2 + 1 dimensions that generalizes the well known Camassa–Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries can be written in terms of five arbitrary constants and three arbitrary functions. Different similarity reductions associated with these symmetries have been derived. Of particular interest are the reduced hierarchies whose 1 + 1 Lax pair is also non-isospectral.

Integrable 1+1 dimensional hierarchies arising from reduction of a non-isospectral problem in 2+1 dimensions

This work presents a classical Lie point symmetry analysis of a two-component, non-isospectral Lax pair of a hierarchy of partial differential equations in

Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in 2 + 1 dimensions

2+1

Jacobi-Lie systems: Fundamentals and low-dimensional classification

dimensions, which can be considered as a modified version of the Camassa-Holm hierarchy in