American Institute of Mathematical Sciences

$\frac{d^2x}{dt^2}+\arctan x=p(t),$

where $p(t+1)=p(t)$ is a smooth function.

i$u_t=u_{x x}-mu-f(\beta t,x)|u|^2 u,$

with the boundary conditions

$u(t,0)=u(t,a\pi)=0, \ -\infty < t < \infty,$

where $m$ is real and $f(\beta t,x)$ is real analytic and quasi-periodic on $t$ satisfying the non-degeneracy condition

$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^Tf(\beta t,x)dt\equiv f_0=$ const., $\quad 0\ne f_0 \in\mathbb R,$

with $\beta\in\mathbb R^b$ a fixed Diophantine vector.