KP-II approximation for a scalar FPU system on a 2D square lattice

TL;DR

This paper demonstrates that the KP-II equation accurately predicts the dynamics of a scalar FPU system on a 2D lattice, extending approximation results to arbitrary wave directions using Fourier analysis in strain variables.

Contribution

It introduces a novel Fourier transform approach in strain variables to validate KP-II approximation for scalar FPU systems in any propagation direction.

Findings

KP-II equation predicts scalar FPU dynamics accurately

Extension of approximation results to arbitrary wave directions

Use of Fourier transform in strain variables enhances analysis

Abstract

We consider a scalar Fermi-Pasta-Ulam (FPU) system on a square 2D lattice. The Kadomtsev-Petviashvili (KP-II) equation can be derived by means of multiple scale expansions to describe unidirectional long waves of small amplitude with slowly varying transverse modulations. We show that the KP-II approximation makes correct predictions about the dynamics of the original scalar FPU system. An existing approximation result is extended to an arbitrary direction of wave propagation. The main novelty of this work is the use of Fourier transform in the analysis of the FPU system in strain variables.