One-component delocalized nonlinear vibrational modes of square lattices

All possible one-component delocalized nonlinear vibrational modes (DNVMs) in a square lattice are analyzed. DNVMs are obtained taking into account exclusively the symmetry of a square lattice, and therefore, they exist regardless of the type of interactions between particles. In this work, the interactions of the nearest and next nearest neighbors are described by the beta-FPUT potential. For each DNVM, frequency, kinetic and potential energies are described as functions of amplitude. The mechanical stresses caused by DNVMs and the effect of DNVMs on the stiffness constants of the lattice are presented. DNVMs with higher vibration frequencies have a stronger effect on the mechanical properties of the lattice. Examples of analytical analysis of DNVMs are given. It is found that two of the sixteen one-component DNVMs can have frequencies above the phonon band in the entire range of their amplitudes. It is shown that these DNVMs can be used to construct discrete breathers by applying localizing functions. The modulational instability of these two DNVMs leads to the formation of chaotic DBs. The presented results contribute to a better understanding of the nonlinear dynamics of a square lattice by analyzing the properties of a class of delocalized exact solutions and demonstrating their connection with discrete breathers.

Automated summary

This paper analyzes all possible one-component delocalized nonlinear vibrational modes (DNVMs) in a square lattice. The DNVMs are obtained solely based on the symmetry of the lattice, thus existing regardless of the particle interactions. The interactions between nearest and next nearest neighbors are described using the beta-FPUT potential. Frequency, kinetic and potential energies of each DNVM are described as functions of amplitude. The paper also presents mechanical stresses caused by DNVMs and the effect of DNVMs on the lattice's stiffness constants. DNVMs with higher vibration frequencies have a stronger impact on the lattice's mechanical properties. Analytical analysis of DNVMs is provided. It is discovered that two out of the sixteen one-component DNVMs can have frequencies above the phonon band across their entire amplitude range. These DNVMs can be used to construct discrete breathers by applying localizing functions. The modulational instability of these two DNVMs leads to the formation of chaotic DBs. The presented results contribute to a better understanding of the nonlinear dynamics of a square lattice by analyzing the properties of a class of delocalized exact solutions and demonstrating their connection with discrete breathers.