A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions

In this paper, one-dimensional (1D) nonlinear Schrödinger equationiut-uxx+mu+∂g(u,u¯)∂u¯=0,with Periodic Boundary Conditions is considered; m∉112Z is a real parameter and the nonlinearityg(u,u¯)=∑j,l,j+l⩾4ajluju¯l,ajl=alj∈R,a22≠0is a real analytic function in a neighborhood of the origin. The KAM machinery is adapted to fit the above equation so as to construct small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system.