Nonlinear theory of waves in solid state with cardinally changing crystalline structure

A nonlinear theory of propagating periodic and nonlinear solitary waves (like kinks and solitons) related to the motion of defects in crystals and of specific periodic waves into which the former ones transform in the field of the compression stress was developed. The role of intense tension stress leading to the heavy structural rearrangement of the crystal as a result of the effect of the external stress on the interatomic potential barriers was taken into account as well. Crystals with a complex lattice consisting of two sublattices were considered. Arbitrarily large displacements of sublattices were analyzed. The nonlinear theory is based on an additional element of the translational symmetry typical for complex lattices but not introduced earlier in solid-state physics. The variational equations of macroscopic and microscopic displacements turn out to be a nonlinear generalization of the linear equations of acoustic and optical modes obtained by Carman, Born, and Huang Kun. The microscopic displacement fields are described by the nonlinear sine-Gordon equation. In the one-dimensional case, exact solutions of the nonlinear equations were found and their features were revealed. In the case of two-dimensional (2+1) fields, new methods of the exact solutions of the sine-Gordon equation were developed. They describe the interaction of the nonlinear waves with the structural inhomogeneities of solid state due to the external fields of stress and deformations.