Solitonlike states in a one-dimensional nonlinear Schrödinger equation with a deterministic aperiodic potential

We study the electronic properties of a one-dimensional deterministic aperiodic nonlinear lattice using a tight-binding form of the nonlinear Schrödinger equation. A nonlinear eigenvalue problem is solved for self-consistent solutions, and the corresponding transmission problem is studied using a two-dimensional nonlinear mapping. When the aperiodicity is chosen according to the Thue-Morse sequence and the nonlinearity is small, we find solitonlike solutions in a similar way as for a periodic lattice. The coupling of an incoming plane wave to these structures results in peaks of perfect transmission in the transmission gaps of the corresponding linear model. Transmission peaks due to dark solitons, and to combinations of several weakly interacting solitons, so called ‘‘multisolitons’’ or ‘‘soliton trains’’ are also found. For larger nonlinearity we find that bounded and diverging solutions are mixed in an intricate pattern; the set of energy values yielding bounded orbits for a given nonlinearity strength showing a Cantor-like structure reminiscent of the linear Thue-Morse system.