Darboux integrability in more than two dimensions

Abstract

Darboux's method reduces the problem of integrating some partial differential equations in two independent variables to the easier problem of solving systems of ordinary differential equations. Those equations to which it applies share many of the properties of so-called integrable systems. In the 1950's, Parsons extended Darboux's method to a restricted class of equations in three or more independent variables. This paper reviews Parson's approach within the geometrical framework of exterior differential systems, which should facilitate the search for more comprehensive generalisations.