Integrability conditions for differential systems

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold (which one sets equal to 0 to find solutions to the system). Given a collection of differential 1-forms on an -dimensional manifold , an integral manifold is an immersed (not necessarily embedded) submanifold whose tangent space at every point is annihilated by (the pullback of) each . A maximal integral manifold is an immersed (not necessarily embedded) submanifold such that the kernel of the restriction map on forms is spanned by the at every point of . If in addition the are linearly independent, then is ()-dimensional. A Pfaffian system is said to be completely integrable if admits a foliation by maximal integral manifolds. (Note that the foliation need not be regular; i.e. the leaves of the foliation might not be embedded submanifolds.) An integrability condition is a condition on the to guarantee that there will be integral submanifolds of sufficiently high dimension. The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.) Not every Pfaffian system is completely integrable in the Frobenius sense.