Year: 2012
Author: Yixian Gao
Communications in Mathematical Research , Vol. 28 (2012), Iss. 1 : pp. 51–64
Abstract
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 $m ∈ (0, M^∗] ⊂ \boldsymbol{R}$ ($M^∗$ 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2012-CMR-19063
Communications in Mathematical Research , Vol. 28 (2012), Iss. 1 : pp. 51–64
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 14
Keywords: beam equation KAM theorem quasi-periodic solution partial Birkhoff normal form.