Justification of the KP-II approximation in dynamics of two-dimensional FPU systems

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