SEMINARIO DEI DOTTORANDI We consider a class of fully nonlinear, autonomous and reversible Schrödinger equations on the circle and we prove the existence and the stability of Cantor families of (small amplitude) quasi-periodic solutions. The proof is based on a combination of different ideas: (i) we perform a 'weak' Birkhoff normal form step in order to find an approximately invariant manifold on which the dynamics is approximately integrable; (ii) we introduce a suitable generalization of a KAM nonlinear iteration for 'tame' and 'unbounded' vector fields based on the invertibility of the linearized equation in a neighborhood of the origin; (iii) we exploit the 'pseudo-differential structure' of the vector field and we prove the invertibility of the linearized operator, using a regularization procedure which conjugates the operator to a differential operator with constant coefficients plus a bounded remainder. This latter step is obtained through transformations generated by torus diffeomorphisms and pseudo-differential operators. Then we use a KAM-like reducibility scheme that reduces to constant coefficients the linearized operator at the solution. This gives the linear stability.