The nonlinear theory of localized waves in a complex crystalline lattice as a discrete continual model

The nonlinear theory of elastic and nonelastic deformations accompanied with deep reconstruction of an initially ideal lattice (switching of interatomic bonds, changing of the class of symmetry, formation of new phases, singular defects and heterogeneities, fragmentation of the lattice) is presented. The proposed theory is a generalization of the classic linear theory (Carman, Born, Kun Huang) of a complex crystalline lattice consisting of two (and more) sublattices and is achieved by introducing the nonlinear equations of acoustic and optic oscillation modes. These equations are derived based on the new principle of internal translation of the complicated lattice invariance relative to the mutual sublattice translation for one period. As a result, it is possible to reach and overcome bifurcation points, i.e., thresholds of lattice stability under catastrophic deformations. The universal mechanism of these effects consists in lowering the potential barriers due to great external stresses. The formation of defects such as dislocations and disclinations and their propagation as localized waves are considered. Soliton-like and kink-like supersonic and subsonic waves have been found out. Some criteria of their excitation by external stresses have been established.