Generalized Symmetries

Abstract

The symmetry groups of differential equations or variational problems considered so far in this book have all been local transformation groups acting “geometrically” on the space of independent and dependent variables. E. Noether was the first to recognize that one could significantly extend the application of symmetry group methods by including derivatives of the relevant dependent variables in the transformations (or, more correctly, their infinitesimal generators). More recently, these “generalized symmetries”† have proved to be of importance in the study of nonlinear wave equations, where it appears that the possession of an infinite number of such symmetries is a characterizing property of “solvable” equations, such as the Korteweg-de Vries equation, which have “soliton” solutions or can be linearized either directly or via inverse scattering.