Korteweg--de Vries limit for the Fermi--Pasta--Ulam system

TL;DR

This paper demonstrates that solutions of the infinite Fermi--Pasta--Ulam system can be approximated by KdV equation solutions as the lattice spacing tends to zero, using advanced dispersive PDE techniques.

Contribution

It introduces simplified dispersive PDE methods for the infinite FPU system and establishes a KdV limit with reduced hypotheses and regularity requirements.

Findings

Solutions approximate KdV solutions as lattice spacing approaches zero

Simplifies previous hypotheses and reduces regularity requirements

Provides a rigorous link between FPU system and KdV equation

Abstract

In this paper, we develop dispersive PDE techniques for the Fermi--Pasta--Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg--de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study [45].