Thresholds for soliton creation in the Ablowitz–Ladik lattice

Square barrier initial potentials for the Ablowitz–Ladik (AL) lattice are considered, both in the single component as well as in the vector (Manakov) case. We determine the threshold condition for creating solitons with such initial conditions in these integrable, discrete versions of the nonlinear Schrödinger equation for the case of one-, two-, three- and four-site barriers. We find that for square barriers in the scalar case, it is impossible to generate a soliton with a single-site excitation, while only one soliton can be produced from a two-site and three-site square barrier. Finally, in the four-site case, there appear to be two thresholds, one leading to a soliton and a second one to a breathing soliton. We illustrate the differences of the vector case from the scalar one for initial conditions with disjoint support between the components, and also discuss the case of non-square barriers. The analytical findings are corroborated by numerical simulations in all the presented cases.