A Computer Algebra Application to Determination of Lie Symmetries of Partial
Differential Equations

Vladimir Pulov
Department of Physics, Technical University-Varna, Varna, Bulgaria

Eddy Chacarov
Department of Mathematics, Varna Free University, Varna, Bulgaria

Ivan Uzunov
Department of Physics, Technical University, Sofia, Bulgaria

Although the procedure of establishing the Lie symmetry transformations leaving
a system of partial differential equations invariant is straightforward, it
often seems formidable to be followed.  This is primarily due to the
difficulties brought about by the great number of symbolic calculations that
must be performed in order to create and solve the so called defining system.
The operation are routine but very tedious to be done by hand even when the
symmetries admitted by a single partial differential equation with the fewest
number of variables are being sought.  The computational problems can be
overcome by using some computer system

for doing symbolic calculations, such as Reduce, MATHEMATICA, Maple, etc.  The
aim of this paper is to present an algorithm of a computer implementation of
the Lie method, showing how the most general Lie group of symmetry point
transformations, admitted by a given system of partial differential equations,
can be obtained in a more advantageous way by utilizing the automatic symbolic
computations of MATHEMATICA.  The algorithm described here, is designed to
create and solve the defining system of an arbitrary number of simultaneous
partial differential equations without any restrictions on the number of the
independent and the dependent variables, as well as on the highest order of the
derivatives that may be involved.  Some results of the package application to
partial differential equations from nonlinear fiber optics are given.