Construction of quasi-periodic solutions for the quintic Schrödinger equation on the two-dimensional torus 𝕋²

Zhang, Min; Si, Jianguo

doi:10.1090/tran/8329

Construction of quasi-periodic solutions for the quintic Schrödinger equation on the two-dimensional torus $\mathbb {T}^2$
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by Min Zhang and Jianguo Si PDF

Trans. Amer. Math. Soc. 374 (2021), 4711-4780 Request permission

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.

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