The nonlocal symmetry of the Drinfeld–Sokolov–Satsuma–Hirota system is obtained from the known Lax pair, and infinitely many nonlocal symmetries are given by introducing the internal parameters. Then the nonlocal symmetry is localized to a prolonged system by introducing suitable auxiliary dependent variables. By applying the classical Lie symmetry method to this prolonged system, two main resu...

We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie dimension three. As an application, we determine index symmetry such metrics and prove that singularities moduli space metrics, up to isometric automorphism are contained in subspace classes with maximal symmetry.

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial diierential equations. Using the introduced deenition of reduction of diierential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equat...

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential eq...

In this thesis, we consider smooth optimal control systems that evolve on Lie groups. Pontryagin’s maximum principle allows us to search for local solutions of the optimal control problem by studying an associated Hamiltonian dynamical system. When the associated Hamiltonian function possess symmetries, we can often study the Hamiltonian system in a vector space whose dimension is lower than th...

Solution by M.C. Nucci (Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy) Lie’s monumental work on transformation groups, [12], [13] and [14], and in particular contact transformations [15], has provided systematic techniques for obtaining exact solutions of differential equations [16]. Many books have been dedicated to this subject and its generalizations (...

General Relativity reduced to two dimensions possesses a large group of symmetries that exchange classical solutions. The associated Lie algebra is known to contain the affine Kac-Moody algebra A (1) 1 and half of a real Witt algebra. In this paper we exhibit the full symmetry under the semi-direct product of Lie(A (1) 1 ) by the Witt algebra Lie(W). Furthermore we exhibit the corresponding hid...

This paper begins by introducing the concept of a quasi-hom-Lie algebra, or simply, a qhl-algebra, which is a natural generalization of hom-Lie algebras introduced in a previous paper [14]. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The nat...

We characterize the family of second order potential differential systems, with n degrees of freedom, via their symmetries. Firstly, we calculate explicitly the equivalence Lie algebra and the weak equivalence Lie algebra. It is shown that the equivalence Lie algebra has the dimension n + 4 + n(n− 1) 2 whereas the weak equivalence Lie algebra is infinitedimensional. The later contains strictly ...

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f , or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated over-determined ‘defining system’ of differential equations. The usual computer classification...