The 1-dimensional Frenkel-Kontorova model was modified and generalized so as to describe topological point defects and dislocations in anisotropic crystals of higher dimensions. The main point was that a substrate periodic potential in the Frenkel-Kontorova model was not to be considered to be a given external spatially-periodic force, but was constructed in a self-consistent manner. That is, such that any disturbance in one of the chains caused a violation of the spatial periodicity in adjacent chains of the crystal. Static and moving soliton (kink and anti-kink) solutions were found numerically for 2- and 3-dimensional anisotropic crystals. The bound states of kink-antikink and kink-kink (antikink-antikink) pairs, and their dynamic properties, were studied. Arrays of soliton states were shown to form dislocations of edge type, and their deformation energy distribution on the crystal lattice was calculated. In order to determine soliton profiles and energy distributions on the lattice, a minimization scheme was used which had been proved to be an effective numerical method for finding solitary-wave solutions in complex systems.

Topological Solitons and Dislocations in Two- and Three-Dimensional Anisotropic Crystals. P.L.Christiansen, A.V.Savin, A.V.Zolotaryuk: Physical Review B, 1998, 57[21], 13564-72