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March, 2010 Threshold solutions for the focusing 3D cubic Schrödinger equation
Thomas Duyckaerts , Svetlana Roudenko
Rev. Mat. Iberoamericana 26(1): 1-56 (March, 2010).


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.


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Thomas Duyckaerts . Svetlana Roudenko . "Threshold solutions for the focusing 3D cubic Schrödinger equation." Rev. Mat. Iberoamericana 26 (1) 1 - 56, March, 2010.


Published: March, 2010
First available in Project Euclid: 16 February 2010

zbMATH: 1195.35276
MathSciNet: MR2662148

Primary: 35B40 , 35P25 , 35Q55

Keywords: Blow-up , nonlinear Schrödinger equation , profile decomposition , scattering

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.26 • No. 1 • March, 2010
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