The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation

In this paper, we construct small amplitude quasi-periodic solutionsfor one dimensional nonlinear Schrödinger equationi$u_t=u_{x x}-mu-f(\beta t,x)|u|^2 u,$ with the boundary conditions$u(t,0)=u(t,a\pi)=0, \ -\infty where $m$ is real and $f(\betat,x)$ is real analytic and quasi-periodic on $t$ satisfying thenon-degeneracy condition$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^Tf(\betat,x)dt\equiv f_0=$ const., $\quad 0\ne f_0 \in\mathbb R,$ with $\beta\in\mathbb R^b$ a fixed Diophantine vector.