A KAM theorem for the Hamiltonian with finite zero normal frequencies and its applications

Yuan Wu; Xiaoping Yuan

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arXiv · 1908.11072

A KAM theorem for the Hamiltonian with finite zero normal frequencies and its applications

Yuan Wu·Xiaoping Yuan

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Abstract

In this paper, we investigate the existence of KAM tori for an infinite dimensional Hamiltonian system with finite number of zero normal frequencies. By constructing a constant quantity we show that, for "most" frequencies in the sense of Lebesgue measure, either if the quantity is zero, there is a KAM tori or if the quantity is not zero, there is no KAM tori in some domain. As application, we show that the nonlinear Schr\"{o}dinger equation with a zero frequency possesses many quasi-periodic solutions.

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Yuan Wu, Xiaoping Yuan. 2019-08-29. A KAM theorem for the Hamiltonian with finite zero normal frequencies and its applications. https://arxiv.org/abs/1908.11072

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