On Vessiot's Theory of Partial Differential Equations: A Geometric Approach to Constructing InfinitesimalSolutions

The object of research presented here is Vessiot'stheory of partial differential equations: for a givendifferential equation one seeks linearisations ofsolutions. To this end, one considers thedifferential equation as a fibred submanifold of ajet bundle of appropriate size and then constructs adistribution of vector fields both tangential to thedifferential equation and contained within thecontact distribution of the jet bundle. Among itssubdistributions those are of interest which aretransversal to the base space of the jet bundle asthey may be adapted to give integral manifolds forthe sought solutions. Vessiot's theory is now basedon a rigorous foundation. The relation betweenVessiot's approach and the crucial notions of theformal theory of differential equations (like formalintegrability and involutivity) is clarified. Thepossible obstructions to involution are deducedexplicitly. Necessary and sufficient conditions forVessiot's approach to succeed are given. Existencetheorems for integral distributions and for flatVessiot connections are proven. The structure of thearising equations is simplified. ImplementingVessiot's algorithm is made possible.