A nonlocal integrable generalization of the Frenkel-Kontorova model of dislocation

Abstract

A new simple nonlocal generalization of the Frenkel--Kontorova model of dislocation in solid body as a type of the nonlocal {sine}-Gordon equation with the generalized interaction term is suggested. Its limit cases, symmetries and exact analytical solutions are obtained. This type of the nonlocal {sine}-Gordon equation is shown to possess one-solitonic solutions which are a nonlocal deformation of the corresponding classical solutions of the {sine}-Gordon equation. Exact analytical solutions of this equation and its Lagrangian integrability and geometrical approach are considered.