Filtering the $ L^2- $critical focusing Schrödinger equation

We study the influence of Szegő projector on the $ L^2- $critical non linear focusing Schrödinger equation, leading to the quintic focusing NLS–Szegő equation on the line

$ \begin{equation*} i\partial_t u + \partial_x^2 u + \Pi(|u|^4 u) = 0, \quad (t, x)\in \mathbb{R}\times \mathbb{R}, \qquad u(0, \cdot) = u_0. \end{equation*} $

It has no Galilean invariance but the momentum $ P(u) = \langle -i\partial_x u, u\rangle_{L^2} $ becomes the $ \dot{H}^{\frac{1}{2}}- $norm. Thus this equation is globally well-posed in $ H^1_+ = \Pi(H^1(\mathbb{R})) $, for every initial datum $ u_0 $. The solution $ L^2- $scatters both forward and backward in time if $ u_0 $ has sufficiently small mass. By using the concentration–compactness principle, we prove the orbital stability of some weak type of the traveling wave : $ u_{\omega, c}(t, x) = e^{i\omega t}Q(x+ct) $, for some $ \omega, c>0 $, where $ Q $ is a ground state associated to Gagliardo–Nirenberg type functional

$ \begin{equation*} I^{(\gamma)}(f) = \frac{\|\partial_x f\|_{L^2}^2\|f\|_{L^2}^{4}+\gamma \langle -i \partial_x f , f\rangle_{L^2}^2 \|f\|_{L^2}^2}{\|f\|_{L^{6}}^{6}}, \qquad \forall f\in H^1_+ \backslash \{0\}, \end{equation*} $

for some $ \gamma\geq 0 $. Its Euler–Lagrange equation is a non local elliptic equation. The ground states are completely classified in the case $ \gamma = 2 $, leading to the actual orbital stability for appropriate traveling waves. As a consequence, the scattering mass threshold of the focusing quintic NLS–Szegő equation is strictly below the mass of ground state associated to the functional $ I^{(0)} $, unlike the recent result by Dodson [8] on the usual quintic focusing non linear Schrödinger equation.