American Institute of Mathematical Sciences

September  2009, 8(5): 1585-1606. doi: 10.3934/cpaa.2009.8.1585

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

 1 Department of Mathematics, Nanjing University, Nanjing 210093, China 2 Department of Mathematics, Nanjing Univerisity, Nanjing 210093

Received  September 2008 Revised  February 2009 Published  April 2009

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.

Citation: Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
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