TY - JOUR
T1 - Quasi-Periodic Solutions of the General Nonlinear Beam Equations
AU - Gao , Yixian
JO - Communications in Mathematical Research 
VL - 1
SP - 51
EP - 64
PY - 2021
DA - 2021/05
SN - 28
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/cmr/19063.html
KW - beam equation, KAM theorem, quasi-periodic solution, partial Birkhoff
normal form.
AB - <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>