Symmetries of differential equations. IV

Abstract

By an application of the geometrical techniques of Lie, Cohen, and Dickson it is shown that a system of differential equations of the form x i(ri = Fi (where ri > 1 for every i = 1,...,n) cannot admit an infinite number of pointlike symmetry vectors. When ri = r for every i = 1,...,n, upper bounds have been computed for the maximum number of independent symmetry vectors that these systems can possess: The upper bounds are given by 2n2 + nr + 2 (when r > 2), and by 2n2 + 4n + 2 (when r = 2). The group of symmetries of x̄(r = 0 (r > 1) has also been computed, and the result obtained shows that when n > 1 and r >2 the number of independent symmetries of these equations does not attain the upper bound 2n2 + nr + 2, which is a common bound for all systems of differential equations of the form x̄(r = ̄(t,x̄,...,x̄(r - 1 ) when r > 2. On the other hand, when r = 2 the first upper bound obtained has been reduced to the value n2 + 4n + 3; this number is equal to the number of independent symmetry vectors of the system x..- = 0, and is also a common bound for all systems of the form x..- = F̄(t,x̄, x.-). © 1983 American Institute of Physics.