Existence of wave trains to mass-in-mass lattices

Abstract

The primary focus of our current research is to investigate wave trains in mass-in-mass (MiM) lattices. Specifically, we introduce an efficient perturbation approach within the framework of variational methods to establish the existence of two distinct periodic waveform functions. These waveform functions correspond to beads and resonators, respectively, for the $\beta$-FPU interaction potential. It is worth noting that this perturbation approach is of significant independent interest and holds promising potential applications in related problems. Furthermore, we rigorously demonstrated the existence of wave trains for the asymptotic quadratic potential under the non-resonance condition by employing a saddle point theorem.