abstract:

We prove that solutions of the cubic nonlinear Schr\"odinger equation on $\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts.

Complementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system we use can be approximated by the full PDE.

A key ingredient in these results is the development of a general methodology for transfering uniform global space-time bounds to suitable Fourier truncations of dispersive PDE models.

As an application, we prove symplectic non-squeezing (in the sense of Gromov) for the cubic NLS on $\Bbb{R}^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic non-squeezing result in a scaling-critical setting.

Finally, we discuss implications of non-squeezing on the nature of scattering.

摘要：

我们证明了在$ \ Bbb {R} ^ 2 $上的三次非线性Schr \“ odinger方程的解可以由有限维哈密顿系统近似地统一在有限的初始数据集上。紧凑的对称性：缩放，平移和伽利略增强。

补充这个近似结果，我们证明了我们使用的有限维哈密顿系统的所有解都可以由完整的PDE近似。

这些结果中的关键要素是开发了一种通用方法，该方法可将统一的全球时空边界转移到分散PDE模型的适当傅里叶截断。

作为一个应用，我们证明了在\\ Bbb {R} ^ 2 $上的三次NLS是辛的非压缩（在Gromov的意义上）。这是无限体积中哈密顿量PDE的第一个辛不压缩结果。它也是缩放关键设置中的第一个无条件辛非压缩结果。

最后，我们讨论了非压缩对散射性质的影响。