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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