A new class of Traveling Solitons for cubic Fractional Nonlinear Schrodinger equations

We consider the one-dimensional cubic fractional nonlinear Schr\"odinger equation $$i\partial_tu-(-\Delta)^\sigma u+|u|^{2}u=0,$$ where $\sigma \in (\frac12,1)$ and the operator $(-\Delta)^\sigma$ is the fractional Laplacian of symbol $|\xi|^{2\sigma}$. Despite of lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form $$u(t,x)=e^{-it(|k|^{2\sigma}-\omega^{2\sigma})}Q_{\omega,k}(x-2t\sigma|k|^{2\sigma-2}k),\quad k\in\mathbb{R},\ \omega>0$$ by a rather involved variational argument.