In the stability and blowup for traveling or standing waves in nonlinear Hamiltonian dispersive equations, the non-degeneracy of the linearization about such a wave is of paramount importance. That is, one must verify the kernel of the second variation of the Hamiltonian is generated by the continuous symmetries of the PDE. The proof of this property can be far from trivial, especially in cases where the dispersion admits a nonlocal description where shooting arguments, Sturm-Liouville theories, and other ODE methods may not be applicable. In this talk, we discuss the non degeneracy and nonlinear orbital stability of antiperiodic traveling wave solutions to a class of defocusing NLS equations with fractional dispersion. Key to our analysis is the development of a ground state theory and oscillation theory for linear periodic, fractional Schrodinger operators with antiperiodic boundary conditions. This is joint work with Kyle Claassen (KU).