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

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.

Keywords

• Infinite dimensional Hamiltonian systems
• KAM theory
• beam equations
• quasi-periodic solutions
• invariant torus