We fully develop the concept of causal symmetry introduced in [19]. A causal symmetry is a transformation of a Lorentzian manifold (V, g) which maps every future-directed vector onto a future-directed vector. We prove that the set of all causal symmetries is not a group under the usual composition operation but a submonoid of the diffeomorphism group of V . Therefore, the infinitesimal generati...

We fully develop the concept of causal symmetry introduced in [21]. A causal symmetry is a transformation of a Lorentzian manifold (V, g) which maps every future-directed vector onto a future-directed vector. We prove that the set of all causal symmetries is not a group under the usual composition operation but a submonoid of the diffeomorphism group of V . Therefore, the infinitesimal generati...

We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau’s theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions...

I give a brief summary of the results reported in [1], in collaboration with G. Amelino-Camelia and F. D’Andrea. I focus on the analysis of the symmetries of κ-Minkowski noncommutative space-time, described in terms of a Weyl map. The commutative-spacetime notion of Lie-algebra symmetries must be replaced by the one of Hopf-algebra symmetries. However, in the Hopf-algebra sense, it is possible ...

We present an extension of the methods of classical Lie group analysis of diierential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the diierential equations, symmetries of classical solutions r...

Q -conditional symmetries of the classical diffusive Lotka–Volterra system in the case of one space variable are completely described and a set of such symmetries in explicit form is constructed. The relevant non-Lie ansatz to reduce the diffusive Lotka–Volterra systems with correctly specified coefficients to ODE systems and examples of new exact solutions are found. A possible biological inte...

We describe three ways of modifying the relativistic Heisenberg algebra first one not linked with quantum symmetries, second and third related with the formalism of quantum groups. The third way is based on the identification of generalized deformed phase space with the semidirect product of two dual Hopf algebras describing quantum group of motions and the corresponding quantum Lie algebra. As...

We apply the theory of Lie point symmetries for study a family partial differential equations which are integrable by hyperbolic reductions method and reduced to members Painlevé transcendents. The main results this that from application similarity transformations provided symmetries, all second-order equations, maximal symmetric can be linearized.

In this paper, we consider a generalized BenjaminBona-Mahony-Burgers equation. Classical symmetries of this equation are considered. The functional forms, for which the BBMB equation can be reduced to ordinary differential equations by classical Lie symmetries, are obtained. A catalogue of symmetry reductions and a catalogue of exact solutions are given. A set of new solitons, kinks, antikinks,...