@Article{CMR-28-1,
author = {Yixian, Gao},
title = {Quasi-Periodic Solutions of the General Nonlinear Beam Equations},
journal = {Communications in Mathematical Research },
year = {2012},
volume = {28},
number = {1},
pages = {51--64},
abstract = {<p style="text-align: justify;">In this paper, one-dimensional (1D) nonlinear beam equations of the form $$u_{tt} − u_{xx} + u_{xxxx} + mu = f(u)$$ with Dirichlet boundary conditions are considered, where the nonlinearity $f$ is an
analytic, odd function and $f(u) = O(u^3)$. It is proved that for all $m ∈ (0, M^∗] ⊂ \boldsymbol{R}$&nbsp; &nbsp; ($M^∗$ is a fixed large number), but a set of small Lebesgue measure, the above equations
admit small-amplitude quasi-periodic solutions corresponding to finite dimensional
invariant tori for an associated infinite dimensional dynamical system. The proof is
based on an infinite dimensional KAM theory and a partial Birkhoff normal form
technique.</p>},
issn = {2707-8523},
doi = {https://doi.org/2012-CMR-19063},
url = {https://global-sci.com/article/81816/quasi-periodic-solutions-of-the-general-nonlinear-beam-equations}
}