Introduction to the theory of bushes of nonlinear normal modes for studying large-amplitude atomic vibrations in systems with discrete symmetry

Abstract

The research group from the Rostov State University has been developing the theory of bushes of nonlinear normal modes (NNMs) in Hamiltonian systems with discrete symmetry since the late 90s of the last century. Group-theoretical methods for studying large-amplitude atomic vibrations in molecular and crystal structures were developed. Each bush represents a certain collection of vibrational modes, which do not change in time despite the time evolution of these modes, and the energy of the initial excitation remains trapped in the bush. Any bush is characterized by its symmetry group, which is a subgroup of the system's symmetry group. The modes contained in the given bush are determined by symmetry-related methods and do not depend on the interatomic interactions in the considered system. The irreducible representations of the point and space groups are essentially used in the theory of the bushes of NNMs, and this theory can be considered as a generalization of the well-known Wigner classification of the small-amplitude vibrations in molecules and crystals for the case of largeamplitudes vibrations. Since using of the irreducible representations of the symmetry groups can be an obstacle to an initial familiarization with the bush theory, in the present review, we explain the basic concepts of this theory only with the aid of the ordinary normal modes, which is well known from the standard textbooks considering the theory of small atomic vibrations in mechanical systems. Our description is based on the example of plane nonlinear atomic vibrations of a simple square molecule.

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

“…One essential question arises in connection with this process: whether the symmetry of the initial conditions is inherited by dynamical regimes appearing as a result of the integration of nonlinear differential equations? The point is that the existence of bushes of NNMs as some exact solutions of nonlinear equations is dictated by the conservation of the symmetry of the dynamical regimes during their time evolution [9,10,29]. We show in the next section that simple permutation symmetry is not inherited, in contrast to the inversion symmetry, which is preserved during the time evolution of the initially excited dynamical regime.…”

“…Since inversion symmetry is conserved during the time evolution, the antisymmetric mode can exist as an exact dynamical object for an arbitrarily long time. In fact, it is a certain Rosenberg mode (see [29]).…”

One-component delocalized nonlinear vibrational modes of square lattices

“…Group-theoretical methods for finding subgroups of space groups have been described in 59 , and an elementary introduction to the problems under discussion can be found in 60 .…”

One-component delocalized nonlinear vibrational modes of square lattices

“…in [52], and an elementary introduction to the problems under discussion can be found in [53]. * dryabov@yandex.ru † gchechin@gmail.com ‡ jjjjenia@mail.ru § sash-alex@yandex.ru ¶ elena.a.korznikova@gmail.com * * dmitriev.sergey.v@gmail.com…”