  Dynamics of the Fermi-Pasta-Ulam (FPU) system on a two-dimensional square
lattice is considered in the limit of small-amplitude long-scale waves with
slow transverse modulations. In the absence of transverse modulations, dynamics
of such waves, even at an oblique angle with respect to the square lattice, is
known to be described by the Korteweg-de Vries (KdV) equation. For the three
basic directions (horizontal, vertical, and diagonal), we prove that the
modulated waves are well described by the Kadomtsev-Petviashvili (KP-II)
equation. The result was expected long ago but proving rigorous bounds on the
approximation error turns out to be complicated due to the nonlocal terms of
the KP-II equation and the vector structure of the FPU systems on
two-dimensional lattices. We have obtained these error bounds by extending the
local well-posedness result for the KP-II equation in Sobolev spaces and by
controlling the error terms with energy estimates. The bounds are useful in the
analysis of transverse stability of solitary and periodic waves in
two-dimensional FPU systems due to many results available for the KP-II
equation.
