论文标题
关于2D立方谐振系统的散射渐近学
On scattering asymptotics for the 2D cubic resonant system
论文作者
论文摘要
在本文中，我们证明了2D（离散维度）立方谐振系统的散射渐近学。在Zhao \ cite {z1}中使用了此散射结果，作为在$ \ mathbb {r}^2 \ times \ times \ mathbb {t}^2 $ in $ h^1 $ space上获得立方NLS的散射。此外，在Yang-Zhao \ cite {yz}中证明了1D模拟。尽管该方案也基于Dodson \ cite {d}紧密，但2D情况更为复杂，这会造成一些新的困难。一个障碍是2D中的立方共振的“ $ l^2 $ estimate”失败（我们在本文中也讨论了它，可能具有其自身的利益）。为了解决此问题，我们建立了较弱的估计，并利用了共振系统的对称性，以修改\ cite {yz}的证明。最后，我们在“ WaveGuides上NLS的长时间动态”的研究线上发表了一些评论。
In this paper, we prove scattering asymptotics for the 2D (discrete dimension) cubic resonant system. This scattering result was used in Zhao \cite{Z1} as an assumption to obtain the scattering for cubic NLS on $\mathbb{R}^2\times \mathbb{T}^2$ in $H^1$ space. Moreover, the 1D analogue is proved in Yang-Zhao \cite{YZ}. Though the scheme is also tightly based on Dodson \cite{D}, the 2D case is more complicated which causes some new difficulties. One obstacle is the failure of `$l^2$-estimate' for the cubic resonances in 2D (we also discuss it in this paper, which may have its own interests). To fix this problem, we establish weaker estimates and exploit the symmetries of the resonant system to modify the proof of \cite{YZ}. At last, we make a few remarks on the research line of `long time dynamics for NLS on waveguides'.
