The study “Orbital stability of periodic standing waves for the cubic fractional nonlinear Schrödinger equation” conducted by ITU Department of Mathematics member Prof. Dr. Gülçin M. Muslu was published in “Journal of Differential Equations”. This journal is one of the world's leading mathematical journals. It is ranked 20th out of 333 journals in category Quartile Q1 in Mathematics according to Web of Science database.
The fractional nonlinear Schrödinger (fNLS) equation appears in several physical applications such as fluid dynamics, quantum mechanics, in the description of Boson stars and water wave dynamics. In this paper, the existence and orbital stability of the periodic standing wave solutions for the fNLS equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and it is showed that the corresponding real solution is always positive. The orbital stability is proved for some values of the order of the fractional derivatives by combining some tools regarding positive operators, the oscillation theorem for fractional Hill operators and the Vakhitov-Kolokolov condition. The exact periodic wave solutions of the fNLS equation are not known for the fractional derivative s between 0 and 1. They obtain the periodic standing wave solutions of the fNLS equation by using the Petviashvili’s iteration method numerically.
It is also investigated the Vakhitov-Kolokolov condition numerically which cannot be obtained analytically for some values of the order of the fractional derivative. The numerical results successfully fill the gaps left by the theoretical ones.