Approximation of Polyatomic FPU Lattices by KdV Equations

We consider the evolution of small amplitude, long wavelength initial data by a polyatomic Fermi–Pasta–Ulam lattice differential equation whose material properties vary periodically. Using the methods of homogenization theory, we prove rigorous estimates that show that the solution breaks up into the linear superposition of two appropriately scaled and modulated counterpropagating waves, each of which solves a Korteweg–de Vries equation, plus a small error. The estimates are valid over very long time scales.