INVARIANT TORI FOR THE QUINTIC SCHRODINGER EQUATION WITH QUASI-PERIODIC FORCING ON THE TWO-DIMENSIONAL TORUS UNDER PERIODIC BOUNDARY CONDITIONS

This paper focuses on the quintic Schrodinger equation with quasi-periodic forcing on the two dimensional torus under periodic boundary conditions. By utilizing the measure estimation of infinitely many small divisors, for most values of the frequency vectors, the Hamiltonian of the linear part of the equation can be reduced to an autonomous system by a symplectic change of coordinates. By some transformations of coordinates, the Hamiltonian of the equation can be transformed into an angle-dependent block-diagonal normal form, which can be achieved by choosing the appropriate tangential sites. By an abstract KAM theorem, it is proved that the existence of a class of invariant tori implies the existence of a class of small-amplitude quasi-periodic solutions to the equation.