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Volume 28, Issue 1
Quasi-Periodic Solutions of the General Nonlinear Beam Equations

Yixian Gao

Commun. Math. Res., 28 (2012), pp. 51-64.

Published online: 2021-05

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  • 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.

  • Keywords

beam equation, KAM theorem, quasi-periodic solution, partial Birkhoff normal form.

  • AMS Subject Headings

37K55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-28-51, author = {Gao , Yixian}, title = {Quasi-Periodic Solutions of the General Nonlinear Beam Equations}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {28}, number = {1}, pages = {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.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19063.html} }
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 -

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.

Gao , Yixian. (2021). Quasi-Periodic Solutions of the General Nonlinear Beam Equations. Communications in Mathematical Research . 28 (1). 51-64. doi:
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