Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability

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 $ u_t +u_x +$uux$ +$uxxx$=0, \quad u(x,0) =\phi (x), \qquad $ 0 0, (*) $ u(0,t) =h (t), \qquad u(1,t)=0, \qquad u_x (1,t) =0, \quad 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 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.