Quasi-periodic solutions for beam equations with the nonlinear terms depending on the space variable. Issue 12 (9th September 2020)

• Record Type: Journal Article
• Authors: Wang, Yi; Si, Jianguo
• Is Part Of: Applicable analysis. Volume 99:Issue 12(2020)
• Page Start: 2150
• Page End: 2169
• Publication Date: 2020-09-09
• Subjects: Rolando Magnanini; infinite-dimensional Hamiltonian system -- KAM theory -- beam equation -- x-dependent -- quasi-periodic solution -- normal form; 37K55 -- 70K43 -- 70K45; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2018.1555325
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: This article is devoted to the study of a beam equation with an x -dependent nonlinear term. We construct an analytic and symplectic transformation which changes the Hamiltonian to its Birkhoff normal form. However, the infinitely many coefficients of the Hamiltonian generating this transformation have small denominators. We prove that these denominators do not vanish for all indices and the transformation is canonical. Applying the normal form to a KAM theorem, it is proved that the equation admits quasi-periodic solutions with prescribed frequencies for any fixed potential constant.