Inverse spectral transform for vector fields and dispersionless PDEs

We consider some rigorous aspects of the direct and inverse spectral problems for vector fields. We also present a method to construct solvable nonlinear Riemann inverse problems corrisponding to distinguished implicit solutions of the associated nonlinear PDEs. We discuss some special examples of solvable Riemann problems and the associated wave breaking for the dispersionless Kadomtsev – Petviashvili (dKP) equation. If time remains, we will show how it is possible to solve the Cauchy problem, for small initial data, of the multidimensional non integrable generalization of the dKP equation, deriving explicit formulas for the wave breaking of the solutions. (Joint works with P. G. Grinevich and S. V. Manakov)