The classical problem of water waves on an incompressible irrotational flow is considered. By introducing an appropriate non-dimensionalization, we derive four Korteweg–de Vries equations: two expressed in Cartesian co-ordinates and two in plane polars. The equations are: the classical (plane) KdV equation, the two-dimensional ‘nearly-plane’ equation, the concentric equation and a new ‘nearly-concentric’ equation. On the basis of the underlying water-wave equations, it is seen that two simple transformations exist between these KdV equations.

By constructing appropriate asymptotic regions defined in terms of the relevant small parameters, we show how various initial value problems give rise to certain solutions of the KdV equations. In particular, the generation of the similarity solutions is examined in detail and it is found that these solutions must eventually match to a solution of the full water-wave equations in a neighbourhood of the origin.