Dark solitons in discrete lattices

We analyze the effect of discreteness on properties and propagation dynamics of dark solitons in the discrete nonlinear Schrödinger 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.