Local well-posedness for the quasi-linear Hamiltonian Schr\"odinger equation on tori

We prove a local in time well-posedness result for quasi-linear Hamiltonian Schr"odinger equations on ${\mathbb{T}}^d$ for any $d\ge 1$. For any initial condition in the Sobolev space $H^s$, with $s$ large, we prove the existence and unicity 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.