Extended KdV equation for the case of uneven bottom

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 equation can be obtained only when the bottom is given by a piecewise linear function. For the case of $\alpha=O(\beta)$ and $\delta=O(\beta)$, a unidirectional wave equation is derived in the first order approach with the same limitation for the bottom profile.