Topological solitons and dislocations in two- and three-dimensional anisotropic crystals

The well-known one-dimensional Frenkel-Kontorova model is modified and generalized to describe topological point defects and dislocations in anisotropic crystals of higher dimensions. The main point of our modification is that a substrate periodic potential in the Frenkel-Kontorova model is not considered as a given external spatially periodic force, but it is constructed in a self-consistent manner, such that any disturbance in one of the chains causes a violation of spatial periodicity in the adjacent chains of the crystal. Static and moving soliton (kink and antikink) solutions are found numerically in two- and three-dimensional anisotropic crystals. Bound states of kink-antikink and kink-kink (antikink-antikink) pairs and their dynamical properties are studied. Arrays of soliton states are shown to form dislocations of the edge type and their deformation energy distribution on the crystal lattice is calculated. In finding the soliton profiles and energy distributions on the lattice, we apply the minimization scheme that has proven to be an effective numerical method for seeking solitary wave solutions in complex systems. The collision dynamics of the point defects are also investigated.