Electron. J. Diff. Equ., Vol. 2015 (2015), No. 11, pp. 1-20.

Quasi-periodic solutions of nonlinear beam equations with quintic quasi-periodic nonlinearities

Qiuju Tuo, Jianguo Si

Abstract:
In this article, we consider the one-dimensional nonlinear beam equations with quasi-periodic quintic nonlinearities
$$ u_{tt}+u_{xxxx}+(B+ \varepsilon\phi(t))u^5=0 $$
under periodic boundary conditions, where B is a positive constant, $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in t with frequency vector $\omega=(\omega_1,\omega_2,\dots,\omega_m)$. It is proved that the above equation admits many quasi-periodic solutions by KAM theory and partial Birkhoff normal form.

Submitted October 24, 2014. Published January 7, 2015.
Math Subject Classifications: 35L05, 37K50, 58E05.
Key Words: Infinite dimensional Hamiltonian systems; KAM theory; beam equations; quasi-periodic solutions; invariant torus.

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Qiuju Tuo
School of Mathematics, Shandong University
Jinan, Shandong 250100, China
email: qiujutuo@163.com
Jianguo Si
School of Mathematics, Shandong University
Jinan, Shandong 250100, China
email: sijgmath@sdu.edu.cn

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