Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 11, pp. 1-20.
Title: Quasi-periodic solutions of nonlinear beam equations
with quintic quasi-periodic nonlinearities
Authors: Qiuju Tuo (Shandong Univ., Jinan, Shandong, China)
Jianguo Si (Shandong Univ., Jinan, Shandong, China)
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 07, 2015.
Math Subject Classifications: 35L05, 37K50, 58E05.
Key Words: Infinite dimensional Hamiltonian systems; KAM theory;
beam equations; quasi-periodic solutions; invariant torus.