A non-linear Egorov theorem and Poincar\'e-Birkhoff normal forms for quasi-linear pdes on the circle

In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant coefficient para-differential equation. In the case of Hamiltonian equations we also put the system in Poincar'e-Birkhoff normal forms. We apply this transformation to quasi-linear perturbations of the Schr"odinger and Beam equations, obtaining a long time existence result without requiring any symmetry on the initial data. We also provide the local in time well-posedness for quasi-linear perturbations of the Benjamin-Ono equation.