Quasi-periodic solutions for beam equations with the nonlinear terms depending on the space variable

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.

COMMUNICATED BY:

• Rolando Magnanini

KEYWORDS:

• infinite-dimensional Hamiltonian system
• KAM theory
• beam equation
• x-dependent
• quasi-periodic solution
• normal form

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37K55
• 70K43
• 70K45

Acknowledgements

The author would like to appreciate the referee for his/her professional comments and suggestions which helped to improve this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11601270, 11571201, 11526120).