# What kinds of KdV-type equations are allowed by an uneven bottom

## TL;DR

This paper surveys derivations of KdV-type equations over uneven bottoms, revealing that only piecewise linear profiles yield compatible Boussinesq equations and deriving new generalized KdV-type equations.

## Contribution

It identifies the specific bottom profiles that allow compatible Boussinesq equations and derives new generalized KdV-type equations for these cases.

## Key findings

- Compatibility of Boussinesq equations is restricted to piecewise linear bottoms.
- New generalized KdV-type equations are derived for these bottom profiles.
- The correction function has a universal form across cases.

## Abstract

In this study, 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. Six different cases of such ordering are discussed. Surprisingly, for all these cases the Boussinesq equations can be made compatible only for the particular piecewise linear bottom profiles, and the correction function 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.