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A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations
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Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean
Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability
| 1. | Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, United States |
| 2. | Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025 |
$ u_t +u_x +$uux$ +$uxxx$=0, \quad u(x,0)
=\phi (x), \qquad $ 0 < x < 1, t > 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.
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