We study the focusing 3d cubic NLS equation with data at the mass-energy threshold, namely, when . 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 . In this paper, we first exhibit 3 special solutions: and , where is the ground state, exponentially approach the ground state solution in the positive time direction, has finite time blow up and scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to 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.
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Thomas Duyckaerts, Svetlana Roudenko, Threshold solutions for the focusing 3D cubic Schrödinger equation. Rev. Mat. Iberoam. 26 (2010), no. 1, pp. 1–56DOI 10.4171/RMI/592