Muhammad USMAN;Bingyu ZHANG
Journal of Systems Science and Complexity. 2007, 20(2): 284-292.
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 Korteweg-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 finite domain. Consideration is given to
the initial-boundary-value problem
$$
\left\{ \begin{array}{l} u_t +u_x +uu_x +u_{xxx}=0, \quad u(x,0)
=\phi (x), \qquad 0<x< 1, \ t>0,\\[2mm] u(0,t) =h (t), \qquad u(1,t)=0,
\qquad u_x (1,t) =0, \quad t>0. \end{array} \right. \eqno{(*)}
$$
It is shown that if the boundary forcing $h$ is periodic with
small ampitude, then the small amplitude solution $u$ of $(*)$
becomes eventually time-periodic. Viewing $(*)$ (without the
initial condition) as an infinite-dimensional 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.
