Periodic solutions of non-linear Schrödinger equations: A para-differential approach

40 pages. This is the version of the paper to appear in Analysis and PDEsThis paper is devoted to the construction of periodic solutions of non-linear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash-Moser methods. Our approach avoids this, exploiting the possibility of reducing, through para-differential conjugation, the equation under study to an equivalent form for which periodic solutions may be constructed by a classical iteration scheme