The existence of integrable invariant manifolds of Hamiltonian partial differential equations

In this note, it is shown that some Hamiltonian partial differential equations such as semi-linear Schrodinger equations, semi-linear wave equations and semi-linear beam equations are partially integrable, i.e., they possess integrable invariant manifolds foliated by invariant tori which carry periodic or quasi-periodic solutions. The linear stability of the obtained invariant manifolds is also concluded. The proofs are based on a special invariant property of the considered equations and a symplectic change of variables first observed in [26].