Group-Invariant Solutions of Differential Equations

We introduce the concept of a weak symmetry group of a system of partial differential equations, that generalizes the “nonclassical” method introduced by Bluman and Cole for finding group-invariant solutions to partial differential equations. Given any system of partial differential equations, it is shown how, in principle, to construct group-invariant solutions for any group of transformations by reducing the number of variables in the system. Conversely, every solution of the system can be found using this reduction method with some weak symmetry group.