Reducibility and quasi-periodic solutions for a two dimensional beam equation with quasi-periodic in time potential

Min Zhang; Yi Wang; Yan Li; Min Zhang; Yi Wang; Yan Li

doi:10.3934/math.2021039

AIMS Mathematics

2021, Volume 6, Issue 1: 643-674. doi: 10.3934/math.2021039

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Research article

Reducibility and quasi-periodic solutions for a two dimensional beam equation with quasi-periodic in time potential

Min Zhang ^{1
,
,},
Yi Wang ²,
Yan Li ¹

1.
College of Science, China University of Petroleum, Qingdao, Shandong 266580, P. R. China
2.
School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, P. R. China

Received: 21 September 2020 Accepted: 20 October 2020 Published: 23 October 2020
MSC : 37K55, 70K40

This article is devoted to the study of a two-dimensional $(2D)$ quasi-periodically forced beam equation $ u_{tt}+\Delta^2 u+ \varepsilon\phi(t)(u+{u}^3) = 0, \quad x\in\mathbb{T}^2, \quad t\in\mathbb{R} $ under periodic boundary conditions, where $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega = (\omega_1, \omega_2 \ldots, \omega_m)$. We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. By solving the measure estimation of infinitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coefficients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.
- two dimensional beam equation,
- quasi-periodic in time potentials,
- reducibility,
- normal form
Citation: Min Zhang, Yi Wang, Yan Li. Reducibility and quasi-periodic solutions for a two dimensional beam equation with quasi-periodic in time potential[J]. AIMS Mathematics, 2021, 6(1): 643-674. doi: 10.3934/math.2021039

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Abstract

This article is devoted to the study of a two-dimensional $(2D)$ quasi-periodically forced beam equation $ u_{tt}+\Delta^2 u+ \varepsilon\phi(t)(u+{u}^3) = 0, \quad x\in\mathbb{T}^2, \quad t\in\mathbb{R} $ under periodic boundary conditions, where $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega = (\omega_1, \omega_2 \ldots, \omega_m)$. We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. By solving the measure estimation of infinitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coefficients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.

References

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