Jianjun Liu | Akateeminen Kirjakauppa

The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrodinger equation $$sqrt{-1}, u_{t}=u_{xx}-M_{xi}u+varepsilon|u|^2u,$$ subject to Dirichlet boundary conditions $u(t,0)=u(t,pi)=0$, where $M_{xi}$ is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier $M_{xi}$, any solution with the initial datum in the $delta$-neighborhood of a KAM torus still stays in the $2delta$-neighborhood of the KAM torus for a polynomial long time such as $|t|leq delta^{-mathcal{M}}$ for any given $mathcal M$ with $0leq mathcal{M}leq C(varepsilon)$, where $C(varepsilon)$ is a constant depending on $varepsilon$ and $C(varepsilon)rightarrowinfty$ as $varepsilonrightarrow0$.