Lie Algebras of Slow Growth and Klein-Gordon PDE

Abstract

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms: Hence the Lie algebras $\chi (\sinh {u})$ and χ(eu + e− 2u) are slowly linearly growing Lie algebras with average growth rates $\frac {3}{2}$ and $\frac {4}{3}$ respectively.