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. A detailed analysis of the celebrated Korteweg-de Vries equation reveals that this nonlinear evolution equation possesses an infinite dimensional abelian group of symmetry transformations. This group of symmetry transformations is given by the resolvents of the so-called generalized KdV equations. And this striking property is shared by many other nonlinear evolution equations; Only to name a few: Burgers equation, sine-Gordon equation, Zakharov-Shabat equations, Gardner equation etc. Furthermore one discovers that for these equations (except Burgers equation) the structure of this abelian symmetry group is intimately connected with the existence (and description) of multisoliton solutions, and in addition connected to the existence of infinitely many conservation laws (via Noether's theorem or rather a suitable generalization thereof).