  We are interested in the influence of filtering the positive Fourier modes to
the integrable non linear Schr{\"o}dinger equation. Equivalently, we want to
study the effect of dispersion added to the cubic Szeg{\"o} equation, leading
to the NLS-Szeg{\"o} equation on the circle $\mathbb{S}^1$\begin{equation*}i
\partial\_t u + \epsilon^{\alpha}\partial\_x^2 u = \Pi(|u|^2 u),\qquad
0<\epsilon<1, \qquad \alpha\geq 0.\end{equation*}There are two sets of results
in this paper. The first result concerns the long time Sobolev estimates for
small data. The second set of results concerns the orbital stability of plane
wave solutions. Some instability results are also obtained, leading to the wave
turbulence phenomenon.
