Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori

We prove a local in time well-posedness result for quasi-linear Hamiltonian Schrödinger equations on Td for any d≥1. For any initial condition in the Sobolev space Hs, with s large, we prove the existence and uniqueness of classical solutions of the Cauchy problem associated to the equation. The lifespan of such a solution depends only on the size of the initial datum. Moreover we prove the continuity of the solution map.