Frobenius manifold for the dispersionless Kadomtsev-Petviashvili equation

We consider a Frobenius structure associated with the dispersionless Kadomtsev-Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic potential with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.