  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].
