Volume 32, Issue 4
On Reducibility of Beam Equation with Quasi-Periodic Forcing Potential

Commun. Math. Res., 32 (2016), pp. 289-302.

Published online: 2021-05

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• Abstract

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation $u_{tt} + ∆^2u + αu + ϵϕ(t)(u + u^3 ) = 0, α > 0$ in the dimension one is considered, where $u(t, x)$ and $ϕ(t$) are analytic quasi-periodic functions in $t$, and $ϵ$ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.

• Keywords

beam equation, infinite dimension, Hamiltonian system, KAM theory, reducibility.

• AMS Subject Headings

37K55, 35B15

• BibTex
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• TXT
@Article{CMR-32-289, author = {Jing and Chang and and 18566 and and Jing Chang}, title = {On Reducibility of Beam Equation with Quasi-Periodic Forcing Potential}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {32}, number = {4}, pages = {289--302}, abstract = {

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation $u_{tt} + ∆^2u + αu + ϵϕ(t)(u + u^3 ) = 0, α > 0$ in the dimension one is considered, where $u(t, x)$ and $ϕ(t$) are analytic quasi-periodic functions in $t$, and $ϵ$ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2016.04.01}, url = {http://global-sci.org/intro/article_detail/cmr/18901.html} }
TY - JOUR T1 - On Reducibility of Beam Equation with Quasi-Periodic Forcing Potential AU - Chang , Jing JO - Communications in Mathematical Research VL - 4 SP - 289 EP - 302 PY - 2021 DA - 2021/05 SN - 32 DO - http://doi.org/10.13447/j.1674-5647.2016.04.01 UR - https://global-sci.org/intro/article_detail/cmr/18901.html KW - beam equation, infinite dimension, Hamiltonian system, KAM theory, reducibility. AB -

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation $u_{tt} + ∆^2u + αu + ϵϕ(t)(u + u^3 ) = 0, α > 0$ in the dimension one is considered, where $u(t, x)$ and $ϕ(t$) are analytic quasi-periodic functions in $t$, and $ϵ$ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.

Jing Chang. (2021). On Reducibility of Beam Equation with Quasi-Periodic Forcing Potential. Communications in Mathematical Research . 32 (4). 289-302. doi:10.13447/j.1674-5647.2016.04.01
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