Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

Abstract

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-boundary-value problem [Formula: see text] For this problem, it is shown that if the small amplitude, boundary forcing h is periodic of period T, say, then the solution u of (*) is eventually periodic of period T. More precisely, we show for each x > 0, that u(x, t + T) - u(x, t) converges to zero exponentially as t → ∞. Viewing (*) (without the initial condition) as an infinite dimensional dynamical system in the Hilbert space H s (R + ) for suitable values of s, we also demonstrate that for a given, small amplitude time-periodic boundary forcing, the system (*) admits a unique limit cycle, or forced oscillation (a solution of (*) without the initial condition that is exactly periodic of period T). Furthermore, we show that this limit cycle is globally exponentially stable. In other words, it comprises an exponentially stable attractor for the infinite-dimensional dynamical system described by (*).

Locations

• CiteSeer X (The Pennsylvania State University) - View - PDF
• Communications in Contemporary Mathematics - View