The Lie Algebra Structure of Nonlinear Evolution Equations Admitting Infinite Dimensional Abelian Symmetry Groups

Abstract

Hereditary operators in Lie algebras are investigated. These are operators which are characterized by a special algebraic equation and their main property is that they generate abelian subalgebras of the given Lie algebra. These abelian subalgebras are infinite dimensional if the hereditary operator is not cyclic. As a consequence hereditary operators generate on a systematic level nonlinear dynamical systems which possess infinite dimensional abelian groups of symmetry transformations. We show that hereditary operators can be understood as special Lie algebra deformations with a linear interpolation property. In order to construct new hereditary operators out of given ones we study the permanence properties of these operators; this study of permanence properties leads in a natural way to a notion of compatibility. For local hereditary operators it is shown that eigenvector decompositions are time invariant (such an eigenvector decomposition is known to characterize pure multisoliton solutions). Apart from the well-known equations (KdV, sine-Gordon, etc.), we give –as examples– many new nonlinear equations with infinite dimensional groups of symmetry transformations.