Document Type

Article

Publication Date

12-2009

Publication Source

Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Abstract

It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Kortewg-de Vries (KdV) equation and the damped Benjamin-Bona-Mahony (BBM) equation. In this paper we intend to show the same results hold for the pure KdV equation (without the damping terms) posed on a bounded domain. Consideration is given to the initial-boundary-value problem

uuxuxxx 0 < x < 1, t > 0, (*)

It is shown that if the boundary forcing is periodic with small amplitude, then the small amplitude solution of (*) becomes eventually time-periodic. Viewing (*) (without the initial condition ) as an infinite-dimensional dynamical system in the Hilbert space , we also demonstrate that for a given periodic boundary forcing with small amplitude, the system (*) admits a (locally) unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations.

Inclusive pages

1509 - 1523

ISBN/ISSN

1078-0947

Document Version

Postprint

Comments

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems-Series A, following peer review. The publisher-authenticated version is available online.

Permission documentation is on file.

Publisher

American Institute of Mathematical Sciences

Volume

26

Issue

4

Peer Reviewed

yes

Keywords

Forced oscillation, time-periodic solution, the KdV equation, the BBM equation, stability

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