Symmetries and symmetry-invariant solutions of differential equations

Abstract

One of the typical applications of symmetry methods in the study of differential equations is the searching for symmetry-invariant solutions. I present here a review of some of the principal techniques related to this idea, together with a comparison between the various approaches, paying special attention to the notion of conditional symmetry (with a careful distinction between some different definitions),and to the concept of λ-symmetry.The close relationship between λ-symmetries for PDE's and standard symmetries is also pointed out.