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A dualPetrovGalerkin method for two integrable fifthorder KdV type equations
Forced oscillations of the Kortewegde Vries equation on a bounded domain and their stability
1.  Department of Mathematics, University of Dayton, Dayton, OH 454692316, United States 
2.  Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 452210025 
$ u_t +u_x +$uu_{x}$ +$u_{xxx}$=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 timeperiodic. Viewing (*) (without the initial condition ) as an infinitedimensional 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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