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

TL;DR

This paper rigorously justifies the use of the KP-II equation to approximate the dynamics of small-amplitude, long-scale waves in two-dimensional FPU systems, extending previous results to include error bounds and stability analysis.

Contribution

The paper provides the first rigorous proof of KP-II approximation bounds for 2D FPU systems, including error estimates and stability implications.

Findings

KP-II accurately describes wave dynamics in 2D FPU systems

Error bounds for the approximation are established

Results facilitate stability analysis of waves in these systems

Abstract

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.