Existence and stability of standing waves for nonlinear fractional Schroumldinger equations

In this paper, we consider the nonlinear fractional Schrodinger equations. By studying constrained minimization problems and applying the method of concentration-compactness and commutator estimates, we obtain the existence of standing waves for nonlinear fractional Schrodinger equations under some assumptions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4746806]