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Abstract
The paper studies forced surface waves on an incompressible,
inviscid fluid in a two-dimensional channel with a small negative or
oscillatory bump on a rigid flat bottom. Such wave motions are
determined by a non-dimensional wave speed $F$, called Froude
number, and $F=1$ is a critical value of $F$. If $F= 1+ \lambda
\epsilon $ with a small parameter $\epsilon
> 0$, then a forced Korteweg-de Vries (FKdV)
equation can be derived to model the wave motion on the free
surface. In this paper, the case $\lambda > 0$ (or $F> 1$, called
supercritical case) is considered. The steady and unsteady solutions
of the FKdV equation with a negative bump function independent of
time are first studied both theoretically and numerically. It is
shown that there are five steady solutions and only one of them,
which exists for all $\lambda > 0$, is stable. Then, solutions of
the FKdV equation with an oscillatory bump function posed on $R$ or
a finite interval are considered. The corresponding linear problems
are solved explicitly and the solutions are rigorously shown to be
eventually periodic as time goes to infinity, while a similar result
holds for the nonlinear problem posed on a finite interval with
small initial data and forcing functions. The nonlinear solutions
with zero initial data for any forcing functions in the real line
$R$ or large forcing functions in a finite interval are obtained
numerically. It is shown numerically that the solutions will become
eventually periodic in time for a small forcing function. The
behavior of the solutions becomes quite irregular as time goes to
infinity, if the forcing function is large.
Mathematics Subject Classification: Primary: 76B15; Secondary: 76B25.
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