Dark solitons in discrete lattices.

Abstract

We analyze the effect of discreteness on properties and propagation dynamics of dark solitons in the discrete nonlinear Schr\"odinger equation. We show that for small-amplitude nonlinear waves the lattice discreteness induces novel properties of dark solitons, e.g., such solitons may be transformed into brightlike dark solitons on a modulationally stable background. For large-amplitude dark solitons we demonstrate that discreteness effects may be understood as arising from an effective periodic potential to the soliton's coordinate similar to the Peierls-Nabarro (PN) periodic potential for (topological) kinks in the Frenkel-Kontorova model. We calculate the PN barrier (the height of the PN potential) to a dark soliton numerically and, in the case of strong interparticle coupling, also analytically, and discuss how the existence of the PN barrier may affect the mobility of dark solitons in a discrete lattice. In particular, we predict unexpected types of discreteness-induced instabilities for soliton-bearing models showing that, even being at a bottom of the PN potential well, the dark soliton is unstable and it always starts to move after a series of oscillations around the potential minimum. An intuitive picture for such a discreteness-induced nonlinear instability of dark solitons is presented, and the novelty of this phenomenon in comparison to bright solitons is emphasized.