  Reductions of the KP-Whitham system, namely the (2+1)-dimensional
hydrodynamic system of five equations that describes the slow modulations of
periodic solutions of the Kadomtsev-Petviashvili (KP) equation, are studied.
Specifically, the soliton and harmonic wave limits of the KP-Whitham system are
considered, which give rise in each case to a four-component (2+1)-dimensional
hydrodynamic system. It is shown that a suitable change of dependent variables
splits the resulting four-component systems into two parts: (i) a decoupled,
independent two-component system comprised of the dispersionless KP equation,
(ii) an auxiliary, two-component system coupled to the mean flow equations,
which describes either the evolution of a linear wave or a soliton propagating
on top of the mean flow. The integrability of both four-component systems is
then demonstrated by applying the Haantjes tensor test as well as the method of
hydrodynamic reductions. Various exact reductions of these systems are then
presented that correspond to concrete physical scenarios.
