  Two-dimensional reductions of the KP-Whitham system, namely the
overdetermined Whitham modulation system for five dependent variables that
describe the periodic solutions of the Kadomtsev-Petviashvili equation, are
studied and characterized. Three different reductions are considered
corresponding to modulations that are independent of x , independent of y , and
of t (i.e., stationary), respectively. Each of these reductions still describes
dynamic, two-dimensional spatial configurations since the modulated cnoidal
wave generically has a nonzero speed and a nonzero slope in the x y plane. In
all three of these reductions, the properties of the resulting systems of
equations are studied. It is shown that the resulting reduced system is not
integrable unless one enforces the compatibility of the system with all
conservation of waves equations (or considers a reduction to the harmonic or
soliton limit). In all cases, compatibility with conservation of waves yields a
reduction in the number of dependent variables to two, three and four,
respectively. As a byproduct of the stationary case, the Whitham modulation
system for the Boussinesq equation is also explicitly obtained.
