Theory of nonlinear waves in solids undergoing strong rearrangements of their crystalline structure

The nonlinear theory of propagation of nonlinear localized waves (like kinks and solitons), connected with the movement of defects in crystals, is developed. It is assumed that crystals possess a complicated lattice consisting of two sub-lattices. The arbitrary large displacements of sub-lattices, are considered. An additional element of translational symmetry is employed in the theory. The element is typical for complicated lattices, however, it has not been introduced in physics of solid state before. It is evident that the relative displacement of sub-lattices for one period (or for an integer of periods) to a superposition of the sub-lattice with itself reproduces the structure of a complicated lattice. This means that the complicated lattice energy should be a periodic function of the relative rigid displacement of sub-lattices, invariant to such a translation. The variational equations of macro- and micro-displacements turned out to be a nonlinear generalization of the linear equations of acoustic and optical modes obtained by Karman, Born and Huang. Some exact solutions are obtained for the one-dimensional case, and their specific features are revealed.