Conservation laws for a class of Third order Evolutionary differential systems

Abstract

The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation law has a unique representative. The structure of this subspace naturally gives rise to the convenient notion of the weight of a conservation law, and we also compute a universal integrability condition to admit any higher order (weight) conservation law. As an example, the differential system descrbing the flow of the curve in the plane by the derivative of its curvature with respect to the arclength is shown to have the KdV property, i.e., an infinite sequence of conservation laws of distnct weights.