On the relevance of the Rosenau-type equations for nonlinear wave propagation in one-dimensional lattices

Official URL: https://dx.doi.org/10.1080/00036811.2026.2646611

We examine the relevance of the Rosenau-type equations for dispersive elastic waves in one-dimensional nonlinear monatomic mass-spring lattices in the long wave limit. We begin by formal asymptotic derivations of the Rosenau-type unidirectional wave equations from the improved Boussinesq-type equations governing bidirectional wave propagation in the continuum limit. We then rigorously prove that the unidirectional solutions of the improved Boussinesq-type equations are well approximated by the solutions of the Rosenau-type equations over a long time scale. We give the error estimates in terms of two independent, small, positive parameters measuring nonlinear and dispersive effects. Finally, we compare numerically the dynamics of the Rosenau-type equations to that of the improved Boussinesq-type equations for the same initial data. Our theoretical results are quantitatively confirmed with numerical simulation results.