Threshold solutions for the focusing $L^2$ -supercritical NLS Equations

We investigate the $L^2$-supercritical and ${\dot{H}}^1$-subcritical nonlinear Schr"{o}dinger equation in $H^1$. In \cite{G1} and \cite{yuan}, the mass-energy quantity $M(Q{)}^{\frac{1-s_c}{s_c}}E(Q)$ has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present paper, we study the dynamics at the critical level $M(u{)}^{\frac{1-s_c}{s_c}}E(u)=M(Q{)}^{\frac{1-s_c}{s_c}}E(Q)$ and classify the corresponding solutions using modulation theory, non-trivially generalize the results obtained in \cite{holmer3} for the 3D cubic Schr"{o}dinger equation.