The uniqueness and concentration behavior of solutions for a nonlinear fractional Schr\"odinger system

The paper is concerned with a nonlinear system of two coupled fractional Schr\"odinger equations with both attractive intraspecies and attractive interspecies interactions in $\mathbb{R}$. By analyzing an associated $L^2$-constrained minimization problem, the uniqueness of solutions to this system is proved via the implicit function theorem. Under a certain type of trapping potential, by establishing some delicate energy estimates, we present a detailed analysis on the concentration behavior of the solutions as the total strength of intraspecies and interspecies interactions tends to a critical value, where each component of the solutions blows up and concentrates at a flattest common minimum point of the associated trapping potentials. An optimal blow-up rate of solutions to the system is also given.