This book provides an up-to-date (2018) presentation of the shallow water
problem according to a theory which goes beyond the Korteweg-de Vries equation.
When we began studying nonlinear partial dierential equations in 2012,
we were struck by the high number of seemingly miraculous results obtained
within the KdV theory. Yet, realizing that this marvelous theory had been derived
from more general laws of hydrodynamics serving solely as a rst order
perturbative approximation with respect to some small parameters, we were
curious about the consequences of an extension of the perturbative approach
to the next (second) order.
The direct extension of KdV to the second order has been known since
1990 as the extended KdV equation. For short we call it KdV2. This equation
is derived under the same assumptions as KdV and applies to the
As beginners in the eld, we asked whether it was possible to derive an
extended KdV type equation for an uneven bottom according to the same