Forced oscillations of the Kortewegde Vries equation on a bounded domain and their stability doi:10.3934/dcds.2010.26.1509 Abstract Full Text (216.9K) Related Articles
Muhammad Usman  Department of Mathematics, University of Dayton, Dayton, OH 454692316, United States (email) 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 Kortewgde Vries (KdV) equation and the damped BenjaminBonaMahony (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 initialboundaryvalue problem
$ u_t +u_x +$uu_{x}$ +$u_{xxx}$=0, \quad u(x,0)
=\phi (x), \qquad $ 0 < x < 1, t > 0, (*) It is shown that if the boundary forcing $h$ is periodic with small amplitude, then the small amplitude solution $u$ of (*) becomes eventually timeperiodic. Viewing (*) (without the initial condition ) as an infinitedimensional dynamical system in the Hilbert space $L^2(0,1)$, 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.
Keywords: Forced oscillation, timeperiodic solution, the
KdV equation, the BBM equation, stability.
Received: November 2008; Revised: January 2009; Published: December 2009. 
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