Threshold solutions for the focusing 3D cubic Schrödinger equation

Abstract

We study the focusing 3d cubic NLS equation with $H^1$ data at the mass-energy threshold, namely, when $M[u_0]E[u_0]=M[Q]E[Q]$. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when $M[u_0]E[u_0]<M[Q]E[Q]$. In this paper, we first exhibit 3 special solutions: $e^{it}Q$ and $Q^{\pm }$, where $Q$ is the ground state, $Q^{\pm }$ exponentially approach the ground state solution in the positive time direction, $Q^{+}$ has finite time blow up and $Q^{-}$ scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to ${\dot{H}}^{1/2}$ symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.