The new Lie point reductions and similarity solutions for the Gordon-type equation of the Milne metric

In this paper, three similarity solutions of a mathematical model for the gravitational waves of the Milne expanding empty spacetime, namely, the Gordon-type equation of the Milne metric will be determined. This equation is a linear second-order partial differential equation, dependent on a functional parameter. Formerly, some Lie reductions of the equation have been found, only for the special cases of its parameter. In this work, by applying a symmetry method for the Gordon-type equation, its new Lie reductions will be found, in the general form of the parameter. Then, its three similarity solutions will be presented.