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Quasi-Periodic Solutions of Derivative Beam Equation on Flat Tori

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Abstract

In this paper, we consider a class of nonlinear beam equations on flat tori \(\mathbb {T}^d_{\mathfrak {L}}\),

$$\begin{aligned} u_{tt}+\Delta ^2u=\epsilon (-\Delta )^{\alpha } f(\omega t,x,(-\Delta )^{\beta }u), \quad 0<\alpha +\beta \le 1 \end{aligned}$$

and prove that the equation admits many quasi-periodic solutions with the non-resonant frequencies \(\omega \). The main proof is based on an abstract Nash-Moser type implicit function theorem developed in Berti and Bolle (Nonlinearity 25(9):2579–2613, 2012; J Eur Math Soc 15(1):229–286, 2013).

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Acknowledgements

The work is supported by the Postdoctoral Foundation of Jiangsu Province(No. 2021K163B), China Postdoctoral Foundation (No. 2021M692717), and the National Natural Science Foundation of China (No. 12101542).

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  1. School of Mathematical Sciences, Yangzhou University, yangzhou, China

    Yingte Sun

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Sun, Y. Quasi-Periodic Solutions of Derivative Beam Equation on Flat Tori. Qual. Theory Dyn. Syst. 21, 121 (2022). https://doi.org/10.1007/s12346-022-00654-2

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