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Forced oscillations of a class of nonlinear dispersive wave equations and their stability. (English) Zbl 1125.93027
Summary: 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
$\begin{cases} u_t +u_x =uu_x + u_{xxx} =0,\;\;u(x,0)=\phi (x),\quad & 0 < x< 1, t> 0, \\ u(0,t)=h(t),\;\;u(1,t)=0,\;\;u_x (1,t)=0,\quad & t> 0.\end{cases}\tag{*}$ 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.

##### MSC:
 93C10 Nonlinear systems in control theory 93C20 Control/observation systems governed by partial differential equations 35Q53 KdV equations (Korteweg-de Vries equations) 93D99 Stability of control systems 35B10 Periodic solutions to PDEs
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