Can simple KdV-type equations be derived for shallow water problem with bottom bathymetry?

We give a survey of derivations of KdV-type equations with an uneven bottom for several cases when small (perturbation) parameters alpha, beta, delta are of different orders. Besides usual small parameters alpha and beta, determining nonlinearity and dispersion, respectively, the model introduces the third parameter delta, which is related to bottom variations. Six different cases of such ordering are discussed. Surprisingly, for all these cases the resulting Boussinesq equations can be made compatible only for the particular piecewise linear bottom profiles, and the correction term in the final wave equations has a universal form. For such bottom relief, several new KdV-type wave equations are derived. These equations generalize the KdV, the extended KdV (KdV2), the fifth-order KdV (KdV5) and the Gardner equations. Numerical simulations of the solutions to some of these equations are presented and discussed. (C) 2019 The Authors. Published by Elsevier B.V.