Lie systems and selected Lie theory methods in differential equations

04.06.2018 - 07.06.2018 | Warsaw



Tuesday, June 5

Multisymplectic Lie systems: theory, applications, and invariants

X. Gracia
Departament de Matematiques
Universitat Politecnica de Catalunya (Spain)


Based upon physical and mathematical examples, we introduce a new class of Lie systems admitting a compatible multisymplectic structure: the multisymplectic Lie systems. Then, we study the problem of determining a compatible multisymplectic structure for standard Lie systems. In particular, we focus on the so-called locally automorphic Lie systems, namely Lie systems locally diffeomorphic to Lie systems on Lie groups related to a Vessiot--Guldberg Lie algebra of right-invariant vector fields.

Multisymplectic methods to obtain nonlinear superposition rules

J. de Lucas
Department of Mathematical Methods in Physics,
University of Warsaw, Poland


Multisymplectic Lie systems are Lie systems admitting a Lie algebra of (locally) Hamiltonian vector fields relative to a multisymplectic structure. This allows us to attach (locally) every multisymplectic Lie system with a Lie algebra of Hamiltonian differential forms associated with the  Lie algebra of vector fields, a so-called Lie--Hamilton (LH) algebra. In turn, a Lie algebra $\mathfrak{g}$ isomorphic to the LH algebra allows us to construct a co-algebra and a $\mathfrak{g}$-module structure on the tensor algebra $T(\mathfrak{g})$. The invariants of $T(\mathfrak{g})$ relative to its $\mathfrak{g}$-module structure are employed to obtain invariants and constants of motion of the multisymplectic Lie system. The coproduct of the coalgebra is employed to obtain its superposition rules. Results are illustrated with examples of physical and mathematical nature like Schwarz equations.  

Reduction of multisymplectic Lie systems

S. Vilarino
Centro Universitario de la Defensa of Zaragoza,


Let $(N,\Theta,X)$ be a multisymplectic Lie system, namely a triple given by a Lie system $X$ on a manifold $N$ admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic form $\Theta$. Given a common Lie group action of symmetries $\Phi:G\times N\rightarrow N$ of $X$ and $\Theta$, we develop a reduction procedure allowing us to obtain
a new multisymplectic Lie system $(N/G,\bar \Theta,\bar X)$, where $N/G$ is the space of orbits of the Lie group action $\Phi$, the $\bar X$ is the projection of $X$ onto $N/G$, and $\bar \Theta$ is a multisymplectic form univocally determined by $\Theta$ and the Lie group action $\Phi$. Our methods are applied to different types of multisymplectic Lie systems of physical relevance.


Wednesday, June 6

Quasi-Lie schemes: a new technique to integrate PDEs

J. de Lucas
Department of Mathematical Methods in Physics,
University of Warsaw,ul. Pasteura 5, 02-093, Warsaw,


The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining $t$-dependent superposition rules and integrability conditions are analysed. We develop a procedure of constructing quasi-Lie systems through
a generalisation to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess-Zumino-Novikov-Witten models, generalised Abel differential equations, Backlund transformations, as well as other differential equations of physical and mathematical relevance.

Deformations of Lie--Hamilton and Jacobi--Lie systems. A case of  study

D. Wysocki
Department of Mathematical Methods in Physics
University of Warsaw, Poland


Lie--Hamilton systems are Lie systems admitting a Vessiot-Guldberg (VG) Lie algebra, whose elements are Hamiltonian relative to a Poisson bivector. Lie--Hamilton systems describe, as particular cases, t-dependent frequency Winternitz-Smorodinsky oscillators and $t$-dependent frequency harmonic oscillators. Ballesteros et al. recently proposed a Poisson--Hopf deformation procedure of Lie--Hamilton systems [1] that led to interesting physical systems. In this talk, I will describe this procedure and present its generalisation to  Jacobi--Lie systems, i.e. Lie systems with a VG Lie algebra of Hamiltonian vector fields with respect to a Jacobi structure. To illustrate this approach, the Schwarz equation will be discussed.


[1] Ballesteros A., Campoamor-Stursberg R., Fernandez-Saiz E., Herranz J.F., de Lucas J., Poisson-Hopf algebra deformations of Lie-Hamilton systems, J. Phys. A 51, 065202 (2018).


Soliton surfaces obtained via CP^{N−1} sigma models

A.M. Grundland
Centre de Recherches Mathematiques and UQTR, Canada.


This talk is devoted to the study of an invariant formulation of completely integrable CP^{N−1} Euclidean sigma models in two dimensions, defined on the Riemann sphere, having finite actions. Surfaces connected with the CP^{N−1} models, invariant recurrence relations linking the successive projection operators and immersion functions of the surfaces are discussed in detail. We show that the immersion functions of 2D-surfaces associated with the CP^{N−1} model are contained in 2D-spheres in the su(N) algebra. Making use of the fact that the immersion functions of the surfaces satisfy the same Euler-Lagrange equations as the original projector variables, we derive surfaces induced by surfaces and prove that the stacked surfaces coincide with each other, which demonstrates the idempotency of the recurrent procedure. We also demonstrate that the CP^{N−1} model equations admit larger classes of solutions than the ones corresponding to rank-1 Hermitian projectors. This fact allows us to generalize the Weierstrass formula for the immersion of 2D-surfaces in the su(N) algebra and show that in general these surfaces cannot be conformally parametrized. Finally, we consider the connection between the structure of the projective formalism and the possibility of spin representations of the su(N) algebra in quantum mechanics.


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