The formal solution space and formal integrability of non-linear PDE's

In the talk, we explain how to formulate PDE's within the framework of jet spaces. This allows the definition of the so-called formal solution space of a non-linear PDE. In case the PDE is formally integrable, the formal solution space carries in a natural way the structure of a profinite dimensional manifold. We also explain the fundamentals of this particular category of infinite dimensional manifolds, and show that in many ways the profinite dimensional manifolds appearing as formal solution spaces of formally integrable PDE's are easier to deal with than the real solution spaces of these PDE's. In addition, we prove a new criterion for formal integrability formally integrable PDE's and derive from it that the Euler-Lagrange Equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.

The talk is on joint work with Batu Gueneysu, Humboldt University, Berlin.