Abstract:In this paper, we develop an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems. As an application of the theorem, we study the quintic nonlinear Schrödinger equation on the two-dimensional torus \begin{equation*} {\mathrm {i}}u_{t}-\Delta u + {|u|}^4u = 0,\quad x\in \mathbb {T}^2\coloneq \mathbb {R}^2/(2\pi \mathbb {Z})^2,\quad t\in \mathbb {R}. \end{equation*} We obtain a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation. The overall strategy in the proof of the KAM theorem is a normal form techniques sparsing angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. The idea in our proof comes from Geng, Xu, and You [Adv. Math. 226 (2011), pp. 5361–5402], which however has to be substantially developed to deal with the equation above.

---

中文翻译：

---

二维圆环面上五次薛定谔方程的拟周期解的构造 𝕋²

摘要：在本文中，我们为无限维哈密顿系统开发了一个抽象的 KAM（Kolmogorov-Arnold-Moser）定理。作为该定理的应用，我们研究了二维环面上的五次非线性薛定谔方程 \begin{equation*} {\mathrm {i}}u_{t}-\Delta u + {|u|}^4u = 0,\quad x\in \mathbb {T}^2\coloneq \mathbb {R}^2/(2\pi \mathbb {Z})^2,\quad t\in \mathbb {R}。\end{equation*} 我们得到方程的小幅度准周期解的惠特尼平滑族。KAM定理证明的总体策略是一种稀疏角度相关项的范式技术，这可以通过选择合适的切线点来实现。我们证明中的想法来自耿、徐和你 [Adv. 数学。226 (2011)，第 5361-5402 页]，

---