A normal form for beam and non-local nonlinear Schrodinger equations

We discuss normal forms of the completely resonant nonlinear beam equation and nonlinear Schrodinger equation. We work in n > 1 spatial dimensions and study both periodic and Dirichlet boundary conditions on cubes. We discuss the applications to the problem of finding quasi-periodic solutions. In the case of periodic boundary and the dimension n = 2, we apply KAM theory and prove the existence and stability of quasi-periodic solutions.