Wave Propagation in Diatomic Lattices

Abstract

We study periodic traveling waves (wave trains) in diatomic Fermi--Pasta--Ulam chains (FPU). By applying the minimax principle, we demonstrate the existence of two different periodic waveform functions corresponding, respectively, to light and heavy particles. Our approach applies to the FPU $\beta$-model for each wavenumber and each frequency, and to FPU chains with asymptotic quadratic potential for wavenumbers and frequencies satisfying the nonresonance condition. As an application to monatomic lattices, we show for the monatomic soft FPU $\beta$-model the existence of supersonic wave trains with two different waveform functions for adjacent particles, contrary to the nonexistence of supersonic wave trains with only one waveform function.