Extended KdV equation for the case of uneven bottom

Piotr Rozmej, Anna Karczewska

Published 2018-10-16Version 1

We derived consistently, according to the second order perturbation approach, the extended KdV equation for an uneven bottom for the case of $\alpha=O(\beta)$ and $\delta=O(\beta^2)$. This is the only one case when second order KdV type nonlinear wave equation can be derived for arbitrary bounded bottom function. Moreover, we proved that for the case of $\alpha=O(\beta)$ and $\delta=O(\beta)$ a uni-directional wave equation can be obtained neither in first order nor in higher orders when the bottom function.