On the quasi-continuum approximation of some localized patterns in the FPUT lattice

Abstract

In the present work, we present a number of localized wave patterns that are theoretically analyzed and numerically illustrated to be observable within the widely applicable paradigm of the FPUT lattice. In particular, we derive a modified KdV equation from the FPUT lattice, which admits a variety of localized waves including these exact rational solutions representing rogue-wave profiles, solitons and breathers on the top of not only homogeneous, but also periodic elliptic function traveling-wave background. We utilize these exact solutions of the modified KdV reduction to construct consistent initial conditions for the FPUT lattice and perform time stepping of the latter. Relevant comparisons between these numerical solutions of the FPUT lattice and their associated analytical counterparts have been conducted to demonstrate good performance of the derived modified KdV reduction in approximating distinct localized wave structures from the FPUT lattice. This approach paves the way for importing a number of quasi-continuum waveforms to the FPUT lattice and the potential associated physical experiments, including recent ones in mechanical metamaterials.