This paper is mainly a review of the multi–Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negat...

Isotropic Magnetohydrodynamic (MHD) Equilibrium Equations and generalized Anisotropic Magnetohydrodynamic (Chew–Goldberger–Low, CGL) Equilibrium Equations possess infinite-dimensional groups of intrinsic symmetries. We show that certain non-trivial Lie point transformations (that can be obtained by direct application of the general Lie group analysis method) are equivalent to Bogoyavlenskij sym...

We show that all Lie point symmetries of various classes of nonlinear differential equations involving critical nonlinearities are variational/divergence symmetries.

In this paper, by using the nonclassical method, several new symmetries and solutions are obtained, which are unobtainable by Lie classical symmetries.

In this article the symmetries of a class of nonlinear telegraph equations are examined. These equations represent a physical model describing electro magnetic shock waves. The main interest is concentrated on potential symmetries but Lie point and non classical symmetries are also calculated. With the aid of these symmetries explicit new solutions are derived or given implicitly by defining eq...

When we consider a diierential equation = 0 whose set of solutions is S, a Lie-point exact symmetry of this is a Lie-point invertible transformation T such that T (S) = S, i.e. such that any solution to = 0 is tranformed into a (generally, diierent) solution to the same equation; here we deene partial symmetries of = 0 as Lie-point invertible transformations T such that there is a nonempty subs...

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalize...

In this paper we analyze the role of general (possibly nonlinear) time-independent Lie point symmetries in the study of nite-dimensional autonomous dynamical systems, and their relationship with the presence of manifolds invariant under the dynamical ow. We rst show that stable and unstable manifolds are left invariant by all Lie point symmetries admitted by the dynamical system. An identical r...

Lie group theory states that knowledge of a m-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by m the number of equation. We apply this principle by finding dilatations and translations that are Lie point symmetries of considered ordinary differential system. By rewriting original problem in an invariant coordinates set for these symmetri...

The classical reduction of order for scalar ordinary differential equations (ODEs) fails for a system of ODEs. We prove a constructive result for the reduction of order for a system of ODEs that admits a solvable Lie algebra of point symmetries. Applications are given for the case of a system of two second-order ODEs which admits a solvable four-dimensional Lie algebra of point symmetries.