Families of fundamental solitons in the two-dimensional superlattices based on the fractional Schrödinger equation

Abstract

We investigate the existence and stability of fundamental solitons in the optical superlattices with self-defocusing and self-focusing nonlinearity based on the fractional Schrödinger equation. With the self-defocusing nonlinearity, fundamental solitons exist in the first gap, and the stability of solitons is in accordance with the anti-Vakhitov–Kolokolov criterion. However, for the self-focusing nonlinearity, the fundamental solitons exist in the semi-infinite gap, and the stability of solitons is in accordance with the Vakhitov–Kolokolov​ criterion. Moreover, the power, integral form factor, and peak value of fundamental solitons versus propagation constant, Lévy index, and relative strength of superlattice are illustrated. It is necessary to mention that the soliton becomes more and more localized by decreasing the Lévy index in both self-defocusing and self-focusing media. It is due to the fact that small Lévy index leads to weak diffraction. It should be pointed that the solitons can also exist with α<1.