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| We consider the 2D water wave problem in an infinite
long canal of finite depth both with and without surface tension. It
has been proven by several authors that long-wavelength solutions to
this problem can be approximated over a physically relevant timespan
by solutions of the Korteweg-de Vries equation or, for certain
values of the surface tension, by solutions of the Kawahara equation. These proofs
are formulated either in Langrangian or in Eulerian coordinates. In
this talk, we provide an alternative proof, which is simpler, more elementary
and shorter. Moreover, the rigorous justification of the KdV approximation
can be given for the cases with and without surface tension together
by one proof. In our proof, we parametrize the free surface by
arc length and use some geometrically and physically motivated
variables with good regularity properties. |
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