ar X iv : 0 80 2 . 19 54 v 2 [ nl in . S I ] 9 F eb 2 00 9 Symmetry condition in terms of Lie brackets

A passive orthonomic system of PDEs defines a submanifold in the corresponding jet manifold, coordinated by so called parametric derivatives. We restrict the total differential operators and the prolongation of an evolutionary vector field v to this submanifold. We show that the vanishing of their commutators is equivalent to v being a generalized symmetry of the system. 1 Concerning the status of this preprint After writing this preprint, I learnt (from an anonymous referee) that the Liebracket criterion is valid for all systems, not only orthonomic ones. Vinogradovstyle derivation can be found for instance in [5, Chapter 4, §3]. The ‘if’ part of the criterion follows from §3.3, while lemma 3.6 in §3.4 gives the ‘only if’ part. Therefore, the present result is not the most general one. Still the reader may appreciate its Van der Kamp-style derivation. The intrinsic differential operators we will define, can be useful in practical situations. 2 The standard symmetry condition In the majority of cases where exact solutions of differential equations can be found, the underlying property is a (continuous) symmetry of the equation [16, 10]. And, in the theory of integrable equations, the recognition and classification methods based on the existence of symmetries have been particular successful [8, 14, 18, 4, 7]. A symmetry-group transforms one solution of an equation to another solution of the same equation. Although this idea goes back to Sophus Lie, we refer to [10] for a good introduction to the subject, numerous examples, applications and references. And we quote: ’The great power of Lie group theory lies in the crucial observation that one can replace the complicated, nonlinear conditions