ar X iv : 0 80 2 . 19 54 v 1 [ nl in . S I ] 1 4 Fe b 20 08 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 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 [11, 7]. And, in the theory of integrable equations, the recognition and classification methods based on the existence of symmetries have been particular successful [5, 10, 13, 3]. 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 [7] 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 for the invariance of the solution set of an equation under the group transformations by an equivalent linear condition of infinitesimal invariance under the corresponding infinitesimal generators of the group action’ [7]. In this paper we provide a characterization of symmetries that is different from the standard one, generalizing a similar characterization in the special setting of ordinary differential equations [11, eq. (3.35)], and evolution equations [7, Prop. 5.19] to the setting of passive orthonomic systems. The natural framework in which symmetries of differential equations are studied is the so called jet-manifold M . Coordinates on M consist of p independent variables xi, q dependent variables u α and the derivatives of the dependent