  We consider the focusing fractional periodic Korteweg-deVries (fKdV) and
fractional periodic nonlinear Schr\"odinger equations (fNLS) equations, with
$L^2$ sub-critical dispersion. In particular, this covers the case of the
periodic KdV and Benjamin-Ono models. We construct two parameter family of
bell-shaped traveling waves for KdV (standing waves for NLS), which are
constrained minimizers of the Hamiltonian. We show in particular that for each
$\lambda>0$, there is a traveling wave solution to fKdV and fNLS $\varphi:
\|\varphi\|_{L^2[-T,T]}^2=\lambda$, which is non-degenerate and spectrally
stable, as well as orbitally stable. This is done completely rigorously,
without any {\it a priori} assumptions on the smoothness of the waves or the
Lagrange multipliers.
