Construction Of Periodic Solutions Of Nonlinear Beam Equations On T~d By A Para-differential Method

This work is aimed to construct periodic solutions of nonlinear beam equationson the - dimensional tori. For a large set of frequencies, we can show an equivalent form of the above equations by a para-di?erential conjugation. The processes of the construction relies on that for outside a convenient small subset the lower bounded of the eigenvalues of the operator on each “block” and a classical iteration scheme.