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

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  • In this paper, we construct small amplitude quasi-periodic solutions for one dimensional nonlinear Schrödinger equation

    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.

    Mathematics Subject Classification: Primary: 70H08, 70H12; Secondary: 37J40.


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