Related papers: Point transformations in invariant difference sche…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
(2+1) dimensional diffusion equation is considered within the framework of equivalence transformations. Generators for the group are obtained and admissible transformations between linear and nonlinear equations are examined. It is shown…
We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants…
A system is invariant with respect to an input transformation if we can transform any dynamic input by this function and obtain the same output dynamics after adjusting the initial conditions appropriately. Often, the set of all such input…
Using the generalized symmetry method we finish a classification, started in the article [R.N. Garifullin, R.I. Yamilov and D. Levi, Classification of five-point differential-difference equations, J. Phys. A: Math. Theor. 50 (2017) 125201…
We demonstrate that complex point transformations can be used to construct non-Hermitian first integrals, time-dependent Dyson maps and metric operators for non-Hermitian quantum systems. Initially we identify a point transformation as a…
A symmetry classification of possible interactions in a diatomic molecular chain is provided. For nonlinear interactions the group of Lie point transformations, leaving the lattice invariant and taking solutions into solutions, is at most…
Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant…
We investigate the algebras of invariants and the properties of the quotient morphism by an action of a finite group scheme in terms of stabilizers of points.
We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.
The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in other cases up to 10 dimensional. The…
An effective method for generating linear equations of maximal symmetry in their much general normal form is obtained. In the said normal form, the coefficients of the equation are differential functions of the coefficient of the term of…
We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of…
We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the…
Ordinary differential equations (ODEs) and ordinary difference systems (O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective…
A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a…
Critical points of an invariant function may or may not be symmetric. We prove, however, that if a symmetric critical point exists, those adjacent to it are generically symmetry breaking. This mathematical mechanism is shown to carry…
The strict connection between Lie point-symmetries of a dynamical system and its constants of motion is discussed and emphasized, through old and new results. It is shown in particular how the knowledge of a symmetry of a dynamical system…
This paper focuses on flow-adapted point-shifts of point processes on topological groups, which map points of a point process to other points of the point process in a translation invariant way. Foliations and connected components generated…
A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation, by means of its differential representation. The package provides functions which enable us to solve the determining…