The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation
The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation
复制标题
DOI：
10.3934/cpaa.2009.8.1585
复制
发表时间：
2009-04
影响因子：
1
通讯作者：
Lei Jiao;Yiqian Wang
中科院分区：
文献类型：
--
作者：
Lei Jiao;Yiqian Wang
本文构造了一维非线性薛定谔方程I$u_t=u_(Xx)-Mu-f(βt，x)|u|^2 u，$的小振幅拟周期解，其边界条件为：u(t，0)=u(t，a)=0；其中$m$是实数，$f(βt，x)$是$t$上的实解析的拟周期函数，满足非退化条件$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^Tf(\beta t，x)dt等价f_0=$const，$四元f_0 in$mathbb R，$其中$β在R^b$中是一个固定的丢番图向量.
In this paper, we construct small amplitude quasi-periodic solutions for one dimensional nonlinear Schrodinger equation i$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.