2,264 research outputs found
A Necessary Condition for existence of Lie Symmetries in Quasihomogeneous Systems of Ordinary Differential Equations
Lie symmetries for ordinary differential equations are studied. In systems of
ordinary differential equations, there do not always exist non-trivial Lie
symmetries around equilibrium points. We present a necessary condition for
existence of Lie symmetries analytic in the neighbourhood of an equilibrium
point. In addition, this result can be applied to a necessary condition for
existence of a Lie symmetry in quasihomogeneous systems of ordinary
differential equations. With the help of our main theorem, it is proved that
several systems do not possess any analytic Lie symmetries.Comment: 15 pages, no figures, AMSLaTe
Gravitating fluids with Lie symmetries
We analyse the underlying nonlinear partial differential equation which
arises in the study of gravitating flat fluid plates of embedding class one.
Our interest in this equation lies in discussing new solutions that can be
found by means of Lie point symmetries. The method utilised reduces the partial
differential equation to an ordinary differential equation according to the Lie
symmetry admitted. We show that a class of solutions found previously can be
characterised by a particular Lie generator. Several new families of solutions
are found explicitly. In particular we find the relevant ordinary differential
equation for all one-dimensional optimal subgroups; in several cases the
ordinary differential equation can be solved in general. We are in a position
to characterise particular solutions with a linear barotropic equation of
state.Comment: 13 pages, To appear in J. Phys. A: Math. Theo
Lie symmetries of multidimensional difference equations
A method is presented for calculating the Lie point symmetries of a scalar
difference equation on a two-dimensional lattice. The symmetry transformations
act on the equations and on the lattice. They take solutions into solutions and
can be used to perform symmetry reduction. The method generalizes one presented
in a recent publication for the case of ordinary difference equations. In turn,
it can easily be generalized to difference systems involving an arbitrary
number of dependent and independent variables
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