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

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

2011-11-24

arxiv.org/abs/1111.5669v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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